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Cross Sections of High Dimensional Shapes

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Topic: inactiveTopic Cross Sections of High Dimensional Shapes Last updated: 10/9/2016; 10:12:42 PM

userPhilip

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Posted: 3/20/2014; 7:10:44 PM blueArrow

Hello everyone. I just found this neat little program and an incredible use for it. I have been deep in study with deriving midsection cuts of some pretty wild and complex shapes out to the tenth dimension. Up until now, I have been doing them by hand the long way, using a unique equation simplifying notation system. I have learned an algorithm that does all of the math to derive the midsections and merge sequences, when moving through the cut array. After finding this app, I applied the cut equations for all 5 of the 4D hyperdonut shapes, called toratopes in an esoteric terminology. I am an active member on the Higher Dimensions Database forum, where we discuss these shapes in detail. I have learned an incredible amount on there.

And, the results were amazing! I was able to see the cuts and merges rendered in real-time, for the first time ever. Becoming fueled by the instant gratification, I will be applying all of the 5D midsections as well, reduced to 3D. The notation system I use directly converts into equations, which is great news for me. There are many, many shapes to be rendered. Not just hundreds, but thousands! Especially when getting into 8 and 9D. I want to use this program to illustrate the shapes I have come to visualize in my mind, for everyone else. I love to teach about them, but words can only take the non-visual types so far.


userPhilip

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Posted: 4/2/2014; 7:12:29 PM blueArrow

Here's some of what I have done so far:

I can provide the implicit equation for any shape you see, if you are interested. It would be cool to have a drop-down menu to select any hypertorus you wanted.

http://hddb.teamikaria.com/forum/viewtopic.php?f=24&t=801&start=360#p21287

http://hddb.teamikaria.com/forum/viewtopic.php?f=24&t=801&start=360#p21378

http://hddb.teamikaria.com/forum/viewtopic.php?f=24&t=801&start=360#p21387

http://hddb.teamikaria.com/forum/viewtopic.php?f=24&t=801&start=390#p21424

http://hddb.teamikaria.com/forum/viewtopic.php?f=24&t=801&start=390#p21449

http://hddb.teamikaria.com/forum/viewtopic.php?f=24&t=801&start=390#p21524

And many more animations here:

https://www.youtube.com/watch?v=sT-Ob65QRd0

https://www.youtube.com/watch?v=7Io2aiWqgYU

My only request for any potential update, would be for up to seven adjustable parameters. Other than that, this has been an awesome find! My friends and myself have seen some of the most incredible midsections of forbiddingly complicated shapes. It's certainly an awesome way to explore these shapes, and educate people with illustration. I'm nowhere done with this, there's so much to do....


userPhilip

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Posted: 4/4/2014; 10:20:04 PM blueArrow

Made another incredible animation movie and photolog of a 5D shape:

http://hddb.teamikaria.com/forum/viewtopic.php?f=24&t=801&p=21568#p21568

Direct link to animation movie:

https://www.youtube.com/watch?v=wj7c2TnV10I&feature=youtu.be


userPhilip

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Posted: 5/24/2014; 8:58:22 PM blueArrow

Here's another super cool shape I explored recently. It's a 6D innertube called toratigritorus, notated as ((((II)I)(II))I). It's implicit surface equation is:

(sqrt((sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2) - R2)^2 + (sqrt(w^2 + v^2) - R3)^2) - R4)^2 + u^2 - R5^2 = 0

It can be cut down into 3D and explored with seven unique equations:

set bounding box XYZ to -7.5,+7.5

Axial Translation Equations:

a,b,c = 0 ~ 4.5

• ((((I))(I))I) - 4x2 array of 8 torii -- (sqrt((sqrt((sqrt(x^2 + a^2) - 3.75)^2 + b^2) - 1.9)^2 + (sqrt(y^2 + c^2) - 2.25)^2) - 1.75)^2 + z^2 - 1^2 = 0

• ((((II))(I))) - 2 cocirc by 2 conc stacked 2 high of 8 torii -- (sqrt((sqrt((sqrt(x^2 + y^2) - 3.75)^2 + a^2) - 1.9)^2 + (sqrt(z^2 + b^2) - 2.25)^2) - 1.75)^2 + c^2 - 1^2 = 0

• ((((I))(II))) - 2 cocirc stacked 4 high of 8 torii -- (sqrt((sqrt((sqrt(x^2 + a^2) - 3.75)^2 + b^2) - 1.9)^2 + (sqrt(y^2 + z^2) - 2.25)^2) - 1.75)^2 + c^2 - 1^2 = 0

• ((((I)I)(I))) - 2 cocirc in 2x1x2 vert square of 8 torii -- (sqrt((sqrt((sqrt(x^2 + a^2) - 3.75)^2 + y^2) - 1.9)^2 + (sqrt(z^2 + b^2) - 2.25)^2) - 1.75)^2 + c^2 - 1^2 = 0

Rotation Equations

a,b,c = 0 ~ 1.57

• ((((Xy)z)(Yx))Z) - rotation A -- (sqrt((sqrt((sqrt((x*sin(c))^2 + (y*cos(a))^2) - 3.75)^2 + (z*cos(b))^2) - 1.9)^2 + (sqrt((y*sin(a))^2 + (x*cos(c))^2) - 2.25)^2) - 1.75)^2 + (z*sin(b))^2 - 1^2 = 0

• ((((Xz)y)(Yx))Z) - rotation B -- (sqrt((sqrt((sqrt((x*sin(c))^2 + (z*cos(a))^2) - 3.75)^2 + (y*cos(b))^2) - 1.9)^2 + (sqrt((y*sin(b))^2 + (x*cos(c))^2) - 2.25)^2) - 1.75)^2 + (z*sin(a))^2 - 1^2 = 0

• ((((Xy)x)(Yz))Z) - rotation C -- (sqrt((sqrt((sqrt((x*sin(b))^2 + (y*cos(a))^2) - 3.75)^2 + (x*cos(b))^2) - 1.9)^2 + (sqrt((y*sin(a))^2 + (z*cos(c))^2) - 2.25)^2) - 1.75)^2 + (z*sin(c))^2 - 1^2 = 0

And enjoy! I recommend graphing in 30 cubes for exploration, then 90 or 100 for really awesome hi-res structures.

A ton of pics are here:

http://hddb.teamikaria.com/forum/viewtopic.php?f=24&t=801&start=450#p21863


userPhilip

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Posted: 6/2/2014; 2:20:14 AM blueArrow

Here's a nine dimensional hypertorus innertube I explored today. It took several hours just to find a missing pair of parentheses, and resolve the diameters. This thing is freaking crazy, let me tell you....

In the shorthand toratope notation it is defined by :

(((((II)I)(II))I)((II)I))

which converts directly into an equation that defines the surface in nine dimensions:

(sqrt((sqrt((sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2) - R2)^2 + (sqrt(w^2 + v^2) - R3)^2) - R4)^2 + u^2) - R5)^2 + (sqrt((sqrt(t^2 + s^2) - R6)^2 + r^2) - R7)^2 - R8^2 = 0

It has eight diameters dividing up nine dimensions, and is capable of being represented in 3D. We have to make six iterated, dimension-losing midcuts along an axis, to get 9D down to 3D. Each of the six cuts is a direction to translate along or rotate around, in order to trace out the extradimensional missing structure. In its full 9D glory, it's one shape with an unbroken, smoothly curving surface. But in 3D, it's 64 torii in a 4x2x4 array of concentric pairings in the major diameter.

It's 3D midsection notation is (((((I))(I)))((I))), which converts to the equation:

(sqrt((sqrt((sqrt((sqrt(x^2) - 5.5)^2) - 2.75)^2 + (sqrt(y^2) - 3.5)^2) - 3)^2) - 1.9)^2 + (sqrt((sqrt(z^2) - 2.5)^2) - 1.25)^2 - 0.75^2 = 0

A few pics are here: http://hddb.teamikaria.com/forum/viewtopic.php?f=24&t=801&start=540#p22148


userPhilip

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Posted: 6/2/2014; 2:20:21 AM blueArrow

A bunch of exploration scripts for CalcPlot3D are here:

http://hddb.teamikaria.com/forum/viewtopic.php?f=6&t=1940#p21618


userPhilip

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Posted: 6/11/2014; 6:12:39 PM blueArrow

Super-awesome animated GIFs are here:

This links to a post of a newly-rendered 7D shape called Spheritigric Duotorus (((II)I)((II)I)I), scroll down for mind-blowing illustrations of extradimensional rotations,

http://hddb.teamikaria.com/forum/viewtopic.php?f=24&t=801&start=540#p22222


userPhilip

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Posted: 6/13/2014; 2:50:55 PM blueArrow

I've been showcasing the 4 distinct toroids in 4D, check them out.

These are great for beginners, in conceiving the concepts of higher dimensions above 3D. Everything you see in these animations and pictures are directly reflected in their +5D counterparts.

• Torisphere ((III)I)

http://hddb.teamikaria.com/forum/viewtopic.php?f=24&t=801&p=22262#p22255

• Spheritorus ((II)II)

http://hddb.teamikaria.com/forum/viewtopic.php?f=24&t=801&p=22262#p22261

• Tiger ((II)(II))

http://hddb.teamikaria.com/forum/viewtopic.php?f=24&t=801&p=22262#p22262

I'm making Ditorus (((II)I)I) today. It needs six animations to illustrate all possible translations and rotations.


userPhilip

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Posted: 6/19/2014; 12:39:36 AM blueArrow

Lately, I've been showcasing some neat examples of chained 4D shapes. And, illustrating examples of how some cannot be chained. Check them out here:

http://hddb.teamikaria.com/forum/viewtopic.php?f=24&t=1982#p22289

I also learned a new trick with the implicit surface functions. It lets you translate AND rotate the same cut axis, which allows for way more control over steering the 3D hyperplane ( no pun intended ). This allowed me to make some really amazing animated GIFs:

http://hddb.teamikaria.com/forum/viewtopic.php?f=24&t=801&start=570#p22290

And, check out page 21, you will see a very amazing morphing of topology from the new function, in the shape ((((II)I)(II))I) .


userPhilip

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Posted: 6/20/2014; 11:46:01 AM blueArrow

Here's a comprehensive list of all functions I have derived, for all shapes I've explored:

EDIT: Unfortunately, the spacing makes it illegible. I'll try to work on a better looking format to represent them.


userPhilip

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Posted: 6/26/2014; 1:24:19 AM blueArrow

I played with the new function type in a revisit to 6D tiger ditorus ((((II)I)I)(II)) . Made quite a few, each with a unique rotate and translate combination. I also provided the intercept map in all dimensions:

Dimensional Map of ((((II)I)I)(II)) Hyperplane Intercepts -----------------------------------------------------------


XYZWVU 6D Hyperplane

((((II)I)I)(II)) - 1 tiger ditorus

XYZWV 5D Hyperplane

((((I)I)I)(II)) - 2 tiger torii in 2x1x1x1x1 line

((((II))I)(II)) - 2 tiger torii in conc maj pair

((((II)I))(II)) - 2 tiger torii in conc med1 pair

((((II)I)I)(I)) - 2 tritoruses in 1x1x1x1x2 column

XYZW 4D Hyperplane

((((I))I)(II)) - 4 tigers in 4x1x1x1 line

((((I)I))(II)) - 4 tigers in conc maj1 pair in 2x1x1x1 line

((((II)))(II)) - 4 tigers in conc maj1 quartet

((((I)I)I)(I)) - 4 ditoruses in 2x1x1x2 vert square

((((II))I)(I)) - 4 ditoruses in conc maj pair in 1x1x1x2 column

((((II)I))(I)) - 4 ditoruses in conc med pair in 1x1x1x2 column

XYZ 3D Hyperplane

((((I)))(II)) - 8 torii in 1x1x8 column

((((I))I)(I)) - 8 torii in 4x1x2 vert rectangle

((((II)))(I)) - 8 torii in conc maj quartet in 1x1x2 column

((((I)I))(I)) - 8 torii in conc maj pair in 2x1x2 vert square

XY 2D Hyperplane

((((I)))(I)) - 16 circles in 8x2 rectangle

------------------------------


Base Form Equations

• ((((I))I)(I))

(sqrt((sqrt((sqrt(x^2 + 0^2) - 5.75)^2 + 0^2) - 3)^2 + y^2) - 1.5)^2 + (sqrt(z^2 + 0^2) - 2.85)^2 - 0.75^2 = 0

• ((((II)))(I))

(sqrt((sqrt((sqrt(x^2 + y^2) - 5.75)^2 + 0^2) - 3)^2 + 0^2) - 1.5)^2 + (sqrt(z^2 + 0^2) - 2.85)^2 - 0.75^2 = 0

• ((((I)))(II))

(sqrt((sqrt((sqrt(x^2 + 0^2) - 5.75)^2 + 0^2) - 3)^2 + 0^2) - 1.5)^2 + (sqrt(y^2 + z^2) - 2.85)^2 - 0.75^2 = 0

• ((((I)I))(I))

(sqrt((sqrt((sqrt(x^2 + 0^2) - 5.75)^2 + y^2) - 3)^2 + 0^2) - 1.5)^2 + (sqrt(z^2 + 0^2) - 2.85)^2 - 0.75^2 = 0

• (((())I)(II))

(sqrt((sqrt((sqrt(0^2 + 0^2) - 5.75)^2 + 0^2) - 3)^2 + x^2) - 1.5)^2 + (sqrt(y^2 + z^2) - 2.85)^2 - 0.75^2 = 0

----------------------------


Translate + Rotate Functions

• ((((Ac)a)I)(C)) - ((((I))I)(I))

(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + (z*cos(d) - c*sin(d))^2) - 5.75)^2 + (x*cos(b) - a*sin(b))^2) - 3)^2 + y^2) - 1.75)^2 + (sqrt((z*sin(d) + c*cos(d))^2 + 0^2) - 2.85)^2 - 0.75^2 = 0

• ((((A)c)a)(CI)) - ((((I)))(II))

(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + 0^2) - 5.75)^2 + (y*cos(d) - c*sin(d))^2) - 3)^2 + (x*cos(b) - a*sin(b))^2) - 1.5)^2 + (sqrt((y*sin(d) + c*cos(d))^2 + z^2) - 2.85)^2 - 0.75^2 = 0

• ((((Ia))c)(AC)) - ((((I)))(II))

(sqrt((sqrt((sqrt(x^2 + (y*cos(b) - a*sin(b))^2) - 5.75)^2 + 0^2) - 3)^2 + (z*cos(d) - c*sin(d))^2) - 1.5)^2 + (sqrt((y*sin(b) + a*cos(b))^2 + (z*sin(d) + c*cos(d))^2) - 2.85)^2 - 0.75^2 = 0

• ((((I)a)c)(AC)) - ((((I)))(II))

(sqrt((sqrt((sqrt(x^2 + 0^2) - 6)^2 + (y*cos(b) - a*sin(b))^2) - 3)^2 + (z*cos(d) - c*sin(d))^2) - 1.5)^2 + (sqrt((y*sin(b) + a*cos(b))^2 + (z*sin(d) + c*cos(d))^2) - 3)^2 - 0.75^2 = 0

--- a=3 , c=0 / [b,d] go thru [0,0]>[1.57,0]>[1.57,1.57]>[0,1.57]>[0,0] Very interesting topology change!!!

• ((((Ia)c))(AC)) - ((((I)))(II))

(sqrt((sqrt((sqrt(x^2 + (y*cos(b) - a*sin(b))^2) - 6)^2 + 0^2) - 3)^2 + (z*cos(d) - c*sin(d))^2) - 1.5)^2 + (sqrt((y*sin(b) + a*cos(b))^2 + (z*sin(d) + c*cos(d))^2) - 2.85)^2 - 0.75^2 = 0

• ((((A)a)C)(Ic)) - ((((I))I)(I))

(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + 0^2) - 5.75)^2 + (x*cos(b) - a*sin(b))^2) - 3)^2 + (y*sin(d) + c*cos(d))^2) - 1.5)^2 + (sqrt(z^2 + (y*cos(d) - c*sin(d))^2) - 2.85)^2 - 0.75^2 = 0

--- a=5.75 , trans out to single tigritorus, neat rotations

• ((((Ac))C)(Ia)) - ((((I))I)(I))

(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + (y*cos(d) - c*sin(d))^2) - 5.75)^2 + 0^2) - 3)^2 + (y*sin(d) + c*cos(d))^2) - 1.5)^2 + (sqrt(z^2 + (x*cos(b) - a*sin(b))^2) - 2.85)^2 - 0.75^2 = 0

• ((((AC)a))(Ic)) - ((((II)))(I))

(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + (y*sin(d) + c*cos(d))^2) - 5.75)^2 + (x*cos(b) - a*sin(b))^2) - 3)^2 + 0^2) - 1.5)^2 + (sqrt(z^2 + (y*cos(d) - c*sin(d))^2) - 2.85)^2 - 0.75^2 = 0

• ((((AC)a)c)(I)) - ((((II)))(I))

(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + (y*sin(d) + c*cos(d))^2) - 5.75)^2 + (x*cos(b) - a*sin(b))^2) - 3)^2 + (y*cos(d) - c*sin(d))^2) - 1.5)^2 + (sqrt(z^2 + 0^2) - 2.85)^2 - 0.75^2 = 0

• ((((Ac)I)a)(C)) - ((((I)I))(I))

(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + (z*cos(d) - c*sin(d))^2) - 5.75)^2 + y^2) - 3)^2 + (x*cos(b) - a*sin(b))^2) - 1.5)^2 + (sqrt((z*sin(d) + c*cos(d))^2 + 0^2) - 2.85)^2 - 0.75^2 = 0

• ((((Ac)I))(Ca)) - ((((I)I))(I))

(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + (z*cos(d) - c*sin(d))^2) - 5.75)^2 + y^2) - 3)^2 + 0^2) - 1.5)^2 + (sqrt((z*sin(d) + c*cos(d))^2 + (x*cos(b) - a*sin(b))^2) - 2.85)^2 - 0.6^2 = 0

• ((((a))I)(AI)) - (((())I)(II))

(sqrt((sqrt((sqrt((y*cos(b) - a*sin(b))^2 + 0^2) - 5.8125)^2 + 0^2) - 3)^2 + x^2) - 2)^2 + (sqrt((y*sin(b) + a*cos(b))^2 + z^2) - 2)^2 - 0.6^2 = 0

--- set b=0.785 , adj A for 4x OBLQ tiger scan along line

• ((((a))A)(II)) - (((())I)(II))

(sqrt((sqrt((sqrt((x*cos(b) - a*sin(b))^2 + 0^2) - 5.8125)^2 + 0^2) - 3)^2 + (x*sin(b) + a*cos(b))^2) - 2)^2 + (sqrt(y^2 + z^2) - 2)^2 - 0.6^2 = 0

--- set B=0.785 , adj A for 4x tiger dance along line

• ((((a)c)C)(AI)) - (((())I)(II))

(sqrt((sqrt((sqrt((y*cos(d) - c*sin(d))^2 + 0^2) - 5.75)^2 + (x*cos(b) - a*sin(b))^2) - 3)^2 + (x*sin(b) + a*cos(b))^2) - 1.5)^2 + (sqrt((y*sin(d) + c*cos(d))^2 + z^2) - 2.85)^2 - 0.75^2 = 0


userPhilip

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Posted: 7/2/2014; 3:36:01 AM blueArrow

Explored another neat shape today. It's the seven dimensional ((((II)I)(II)I)I) , called the [21-torus 21-tiger 1-torus], or torispheric tigritorus.

Implicit surface equation:

((((II)I)(II)I)I) converts to

(sqrt((sqrt((sqrt(x^2 + y^2) - R1a)^2 + z^2) - R2)^2 + (sqrt(w^2 + v^2) - R1b)^2 + u^2) - R3)^2 + t^2 - R4^2 = 0

X,Y,Z = -8 , +8

Axial Midsection Equations:

• ((((Ii)i)(Ii)i)I) - 8x torii ((II)I) stacked in 4x2 major rectangle array

(sqrt((sqrt((sqrt(x^2 + a^2) - 4)^2 + b^2) - 2)^2 + (sqrt(y^2 + c^2) - 2)^2 + d^2) - 1.75)^2 + z^2 - 1^2 = 0

• ((((Ii)i)(Ii)I)i) - 16x spheres (III) as concentric pair stacked in 4x2 rectangle array

(sqrt((sqrt((sqrt(x^2 + a^2) - 4)^2 + b^2) - 2)^2 + (sqrt(y^2 + c^2) - 2)^2 + z^2) - 1.75)^2 + d^2 - 1^2 = 0

• ((((Ii)i)(II)i)i) - 8x torii ((II)I) as concentric minor pair stacked in 1x1x4 minor column

(sqrt((sqrt((sqrt(x^2 + a^2) - 4)^2 + b^2) - 2)^2 + (sqrt(y^2 + z^2) - 2)^2 + c^2) - 1.75)^2 + d^2 - 1^2 = 0

• ((((Ii)I)(Ii)i)i) - 8x torii ((II)I) as concentric minor pair stacked in 2x1x2 maj/min square array

(sqrt((sqrt((sqrt(x^2 + a^2) - 4)^2 + y^2) - 2)^2 + (sqrt(z^2 + b^2) - 2)^2 + c^2) - 1.75)^2 + d^2 - 1^2 = 0

• ((((II)i)(Ii)i)i) - 8x torii ((II)I) as concentric major/minor pairs stacked in 1x1x2 minor column

(sqrt((sqrt((sqrt(x^2 + y^2) - 4)^2 + a^2) - 2)^2 + (sqrt(z^2 + b^2) - 2)^2 + c^2) - 1.75)^2 + d^2 - 1^2 = 0

Rotation / Rotate + Translate Functions:

• ((((Xy)z)(Yc)x)Z) - ((((I))(I))I)

(sqrt((sqrt((sqrt((x*sin(d))^2 + (y*cos(a))^2) - 4)^2 + (z*cos(b))^2) - 2)^2 + (sqrt((y*sin(a))^2 + c^2) - 2)^2 + (x*cos(d))^2) - 1.25)^2 + (z*sin(b))^2 - 1^2 = 0

• ((((Xz)x)(Yi)y)Z) - ((((I))(I))I)

(sqrt((sqrt((sqrt((x*sin(b))^2 + (z*cos(a))^2) - 4)^2 + (x*cos(b))^2) - 2)^2 + (sqrt((y*sin(d))^2 + c^2) - 2)^2 + (y*cos(d))^2) - 1.25)^2 + (z*sin(a))^2 - 1^2 = 0

• ((((Xz)y)(Yx)Z)i) - ((((I))(I)I)) (sqrt((sqrt((sqrt((x*sin(c))^2 + (z*cos(a))^2) - 4)^2 + (y*cos(b))^2) - 2)^2 + (sqrt((y*sin(b))^2 + (x*cos(c))^2) - 2)^2 + (z*sin(a))^2) - 1.75)^2 + d^2 - 1^2 = 0

• ((((Ai)c)(Ca)i)I) - ((((I))(I))I)

(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + 0^2) - 4)^2 + (y*cos(d) - c*sin(d))^2) - 2)^2 + (sqrt((y*sin(d) + c*cos(d))^2 + (x*cos(b) - a*sin(b))^2) - 2)^2 + 0^2) - 1.75)^2 + z^2 - 1^2 = 0

-- Range: A,C=-8,+8 / B,D=0,1.57 --- B=1.2 , C=0 : animate A, alternate D [A,D] = [-6,0]>[6,0]>[6,1.57]>[-6,1.57]>[-6,0] --- A,C=0 , B=1.2 : animate D [0] > [1.57] > [0]

• ((((Az)y)(Ya)i)Z) - ((((I))(I))I)

(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + (z*cos(d))^2) - 4)^2 + (y*cos(c))^2) - 2)^2 + (sqrt((y*sin(c))^2 + (x*cos(b) - a*sin(b))^2) - 2)^2 + 0^2) - 1.75)^2 + (z*sin(d))^2 - 1^2 = 0

--Range: A=-8,+8 / B,C,D=0,1.57

• ((((Ai)y)(Ya)z)Z) - ((((I))(I))I)

(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + 0^2) - 4)^2 + (y*cos(c))^2) - 2)^2 + (sqrt((y*sin(c))^2 + (x*cos(b) - a*sin(b))^2) - 2)^2 + (z*cos(d))^2) - 1.75)^2 + (z*sin(d))^2 - 1^2 = 0

--Range: A=-8,+8 / B,C,D=0,1.57 --- A=2 : alternate [B,C] = [0,0]>[1.57,0]>[1.57,1.57]>[0,1.57]>[0,0]

• ((((Ad)y)(Ya)i)I) - ((((I))(I))I)

(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + d^2) - 4)^2 + (y*cos(c))^2) - 2)^2 + (sqrt((y*sin(c))^2 + (x*cos(b) - a*sin(b))^2) - 2)^2 + 0^2) - 1.75)^2 + z^2 - 1^2 = 0

--Range: A,D=-8,+8 / B,C=0,1.57 ---A=1.67 / B=1.11 / C=0 : adj D for amazing topology morphs!!

No pictures or animations yet ....

An exploration script for CalcPlot is here:

http://hddb.teamikaria.com/forum/viewtopic.php?f=24&t=801&start=630#p22395

All seven functions above are in it.


userPhilip

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Posted: 7/10/2014; 12:57:20 PM blueArrow

Here's another new shape I've been exploring lately. It's another awesome 7D toroid notated as ((((II)I)((II)I))I), something I call a toratiger duotorus. It can be described as the 6-manifold T2xC2xC2, a " 2-torus bundle over the Clifford 2-manifold, bundled over the Clifford 2-manifold " .

The toratope notation ((((II)I)((II)I))I) converts into an implicit surface equation through the process of:

((((II)I)((II)I))I) = 0

(((II)I)((II)I))I = 0

(((xy)z)((wv)u))t = 0

( ((xy)z) ((wv)u) ) t = 0

( ((x+y)+z) + ((w+v)+u) ) +t = 0

( ((x+y -R1) +z -R2) + ((w+v -R3)+u -R4) -R5) +t -R6 = 0

( ((x+y -R1)^2 +z -R2)^2 + ((w+v -R3)^2 +u -R4)^2 -R5)^2 +t -R6^2 = 0

( ((sqrt(x+y) -R1)^2 +z -R2)^2 + ((w+v -R3)^2 +u -R4)^2 -R5)^2 +t -R6^2 = 0

( (sqrt((sqrt(x+y) -R1)^2 +z) -R2)^2 + ((w+v -R3)^2 +u -R4)^2 -R5)^2 +t -R6^2 = 0

((sqrt((sqrt(x+y) -R1)^2 +z) -R2)^2 + ((sqrt(w+v) -R3)^2 +u -R4)^2 -R5)^2 +t -R6^2 = 0

(sqrt((sqrt((sqrt(x+y) -R1)^2 +z) -R2)^2 + (sqrt((sqrt(w+v) -R3)^2 +u) -R4)^2) -R5)^2 +t -R6^2 = 0

(sqrt((sqrt((sqrt(x^2+y^2)-R1)^2+z^2)-R2)^2+(sqrt((sqrt(w^2+v^2)-R3)^2+u^2)-R4)^2)-R5)^2+t^2-R6^2 = 0

--------------------


Here's the exploration functions for T2xC2xC2 :

7D ((((II)I)((II)I))I) - Toratiger Duotorus , T2xC2xC2 = T2x[T2*T2] , ((((maj1)sec1)((maj2)sec2)tert)min)

(sqrt((sqrt((sqrt(x^2+y^2) -R1a)^2 +z^2) -R2a)^2 + (sqrt((sqrt(w^2+v^2) -R1b)^2 +u^2) -R2b)^2) -R3)^2 +t^2 -R4^2 = 0

• ((((I))((I)))I) - 16 tori in 4x4x1 maj array - Diameter Adjustment Equation

(sqrt((sqrt((sqrt(x^2+0^2) -a)^2 +0^2) -b)^2 + (sqrt((sqrt(y^2+0^2) -a)^2 +0^2) -b)^2) -c)^2 +z^2 -d^2 = 0

• ((((I))((I)))I) - 16 tori in 4x4x1 maj array

(sqrt((sqrt((sqrt(x^2+0^2) -5)^2 +0^2) -2.5)^2 + (sqrt((sqrt(y^2+0^2) -5)^2 +0^2) -2.5)^2) -2)^2 +z^2 -1.25^2 = 0

(sqrt((sqrt((sqrt(x^2+a^2) -5)^2 +b^2) -2.5)^2 + (sqrt((sqrt(y^2+c^2) -5)^2 +d^2) -2.5)^2) -2)^2 +z^2 -1.25^2 = 0

--- XYZ = -12,+12

• ((((Xy)z)((Yc)x))Z) - ((((I))((I)))I)

(sqrt((sqrt((sqrt((x*sin(d))^2+(y*cos(a))^2) -5)^2 +(z*cos(b))^2) -2.5)^2 + (sqrt((sqrt((y*sin(a))^2+c^2) -5)^2 + (x*cos(d))^2) -2.5)^2) -2)^2 +(z*sin(b))^2 -1.25^2 = 0

--- B=1.57/C,D=0 : Adjust A for rotate past 4x diagonal (((II)I)I) in 2x1x4x1 vert rectangle of 8 ditoruses ((((I)I)((I)))I)

• ((((Ac))((C)a))I) - ((((I))((I)))I) --> Explores ((((II)I)((I)))I) , a 1x1x1x4x1 tertiary column of 4x tritoruses

(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2+(y*cos(d) - c*sin(d))^2) -5)^2) -2.5)^2 + (sqrt((sqrt((y*sin(d) + c*cos(d))^2) -5)^2 +(x*cos(b) - a*sin(b))^2) -2.5)^2) -2)^2 +z^2 -1.25^2 = 0

--- [A,B]=[0,1.57] : Animate C=-12.5,+12.5 , Step Adjust D in 6 Incr between 0,1.57

--- [A,B]=[0,1.57] / C=7.5 : Animate D for tritorus empty cut morphing

--- [A,B]=[0,1.57] / C=2.5 : Alternate [B,D] through 4 combos of 0,1.57

Cool animations illustrating this shape are here:

http://hddb.teamikaria.com/forum/viewtopic.php?f=24&t=801&start=660#p22440

and

http://hddb.teamikaria.com/forum/viewtopic.php?f=24&t=801&start=660#p22458


userPhilip

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Posted: 7/17/2014; 11:31:29 AM blueArrow

I've been posting these animations and pics on r/mathpics, and the community has gone wild with it! Check it out:

http://www.reddit.com/r/mathpics/

This is turning into something really great. Paul, I hope we can combine our efforts one day, to possibly refine CalcPlot3D for more adjustable variables, if not more things. Your initial release is awesome, but you probably didn't expect someone like me to come along with my application for it!

I've been pointing a lot of people here, as of late. Your program does an excellent job for exploring multidimensional shapes, come to find out.


userPhilip

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Posted: 10/11/2014; 9:13:12 PM blueArrow

Hey Paul, is there a way to program my computer to take a screenshot for each animation step? I've been thinking about this, lately. For each tick of the slider during animation, the computer takes a screenshot, automatically. It would be very efficient.

Also, here's some awesome animations of a whole ton of 6D and 7D hypertori:

http://hddb.teamikaria.com/forum/viewtopic.php?p=22685#p22685


userPhilip

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Posted: 12/9/2014; 8:35:24 PM blueArrow

It's been a little while since I made any posts here. I'm still exploring multidimensional shapes, of course! In fact, I learned how to define even more shapes now, which are a new addition. My main preference is still the implicit function, which I applied to my notation I created in 2008. Other than toroidal shapes, it defines various flat and/or curve sided prisms. Check it out, my recent breakthrough:

The Construction Operators for Shape M

mI : The Extend/Extrude across N+1 , makes a prism from shape M

m> : The Taper/Scale across N+1 , makes a pyramid or cone from shape M

mO : The Bisecting Rotate into N+1 , makes a sphere, cylinder, cone, or cyclotope from shape M

m(n) : The Fiber Bundle of M over surface of shape N , makes a hypertoric ring or framotope

m[n] : The Cartesian Product with shape M and N , makes (m,n)-duoprism, or other hyperprism not definable with linears

The First Dimension

I - LINE : |x| - a

The Second Dimension

IO - CIRCLE : sqrt(x²+y²) - a²

I> - RIGHT TRIANGLE : ||x|-2y| + |x| - a

II - SQUARE : |x-y| + |x+y| - a

The Third Dimension

IOO - SPHERE : sqrt(x²+y²+z²) - a²

IO(O) - TORUS : (sqrt(x²+y²) - a)² + z² - b²

IO> - CONE : |sqrt(x²+y²) -2z| + sqrt(x²+y²) - a²

IOI - CYLINDER : |sqrt(x²+y²) - z| + |sqrt(x²+y²) + z| - a

I>> - TETRAHEDRON : |||x|-2y|+|x| - 2z| + ||x|-2y|+|x| - a

I>I - TRIANGLE PRISM : |||x|-2y|+|x| - z| + |||x|-2y|+|x| + z| - a

II> - SQUARE PYRAMID : ||x-y|+|x+y| - 3z| + |x-y|+|x+y| - a

III - UNIT CUBE : ||x-y|+|x+y| - 2z| +||x-y|+|x+y| + 2z| - a

The Fourth Dimension

IO[(O)(O)] - TIGER : (sqrt(x²+y²) - a)² + (sqrt(z²+w²) - b)² - c²

IO(O)(O) - DITORUS : (sqrt((sqrt(x²+y²) - a)² + z²) - b)² - c²

IO(OO) - TORISPHERE : (sqrt(x²+y²+z²) - a)² + w² - b²

IOO(O) - SPHERITORUS : (sqrt(x²+y²) - a)² + z² + w² - b²

IOOO - GLOME : sqrt(x²+y²+z²+w²) - a²

IOO> - SPHONE : |sqrt(x²+y²+z²) - 2w| + sqrt(x²+y²+z²) - a²

IOOI - SPHERINDER : |sqrt(x²+y²+z²) - w| + |sqrt(x²+y²+z²) + w| - a

IO>> - DICONE : ||sqrt(x²+y²) -2z| + sqrt(x²+y²) - 2w| + |sqrt(x²+y²) -2z| + sqrt(x²+y²) - a²

IO>I - CONINDER : ||sqrt(x²+y²) -2z| + sqrt(x²+y²) - w| + ||sqrt(x²+y²) -2z| + sqrt(x²+y²) + w| - a

IOIO - DUOCYLINDER : |sqrt(x²+y²) - sqrt(z²+w²)| + |sqrt(x²+y²) + sqrt(z²+w²)| - a²

IIO> - CYLINDRONE : ||sqrt(x²+y²) -z| + |sqrt(x²+y²) +z| - 2w| + |sqrt(x²+y²) -z| + |sqrt(x²+y²) +z| - a

I>>> - PENTACHORON : ||||x|-2y|+|x| -2z| + ||x|-2y|+|x| - 4w| + |||x|-2y|+|x| -2z| + ||x|-2y|+|x| - a

I>>I - TETRAHEDRINDER : ||||x|-2y|+|x|-2z| + ||x|-2y|+|x| - 2w| + ||||x|-2y|+|x|-2z| + ||x|-2y|+|x| + 2w| - a

I>IO - CYLTRIANGLINDER : |||x|-2y|+|x| - sqrt(z²+w²)| + |||x|-2y|+|x| + sqrt(z²+w²)| - a

I>I> - TRIANGLE PRISMID : ||||x|-2y|+|x| -z| + |||x|-2y|+|x| +z| - 2w| + |||x|-2y|+|x| -z| + |||x|-2y|+|x| +z| - a

I>II - TRIANGLE DIPRISM : |||x|-2y|+|x| - |z-w|-|z+w|| + |||x|-2y|+|x| + |z-w|+|z+w|| - a

II>> - DIPYRAMID : |||x-y|+|x+y| -3z| + |x-y|+|x+y| - 3w| + ||x-y|+|x+y| -3z| + |x-y|+|x+y| - a

II>I - PYRAMID PRISM : |||x-y|+|x+y| -3z| + |x-y|+|x+y| - 2w| + |||x-y|+|x+y| -3z| + |x-y|+|x+y| + 2w| - a

IIIO - CUBINDER : ||x-y|+|x+y| - sqrt(z²+w²)| + ||x-y|+|x+y| + sqrt(z²+w²)| - a

III> - CUBE PYRAMID : |||x-y|+|x+y| -2z| + ||x-y|+|x+y| +2z| - 4w| + ||x-y|+|x+y| -2z| + ||x-y|+|x+y| +2z| - a

IIII - TESSERACT : ||x-y|+|x+y| - |z-w|-|z+w|| + ||x-y|+|x+y| + |z-w|+|z+w|| - a

I>[I>] - DUOTRIANGLINDER : |||x|-2y|+|x| - ||z|-2w|-|z|| + |||x|-2y|+|x| + ||z|-2w|+|z|| - a

I'm working on the full 5D list now. I've explored two of them, so far:

II>IO - Cylhemoctahedrinder :

|||x-y|+|x+y|-2z|+|x-y|+|x+y| -3sqrt(w²+v²)| + |||x-y|+|x+y|-2z|+|x-y|+|x+y| +3sqrt(w²+v²)| - a²

3D explore function for II>IO :

abs(abs(abs(x-(y*sin(a)))+abs(x+(y*sin(a))) - 2(z*sin(b))) + abs(x-(y*sin(a)))+abs(x+(y*sin(a))) - 3sqrt((y*cos(a))^2 + (z*cos(b))^2)) + abs(abs(abs(x-(y*sin(a)))+abs(x+(y*sin(a))) - 2(z*sin(b))) + abs(x-(y*sin(a)))+abs(x+(y*sin(a))) + 3sqrt((y*cos(a))^2 + (z*cos(b))^2)) - 3.5^2

---[A,B] Rotate Positions

[0,0] - IIO cylinder

[1.57,0] - III cube

[1.57,1.57] - II> square pyramid

[0,1.57] - I>I triangle prism

Animated GIF : http://hddb.teamikaria.com/forum/viewtopic.php?p=23098#p23098

• IO>[I>] - Contrianglinder :

||sqrt(x²+y²)-2z|+sqrt(x²+y²) - ||w|-2v|-|w|| + ||sqrt(x²+y²)-2z|+sqrt(x²+y²) + ||w|-2v|+|w|| - a

3D explore function for IO>[I>] :

abs(abs(sqrt(x^2+(y*sin(a))^2) - 2(z*sin(b))) + sqrt(x^2+(y*sin(a))^2) - abs(abs((y*cos(a)))-2(z*cos(b))) - abs((y*cos(a)))) + abs(abs(sqrt(x^2+(y*sin(a))^2) - 2(z*sin(b))) + sqrt(x^2+(y*sin(a))^2) + abs(abs((y*cos(a)))-2(z*cos(b))) + abs((y*cos(a)))) - 10

[a,b] - 3D Midsection Positions

[0,0] - Triangle Prism I>I

[1.57,0] - Cylinder IOI

[1.57,1.57] - Cone IO>

[0,1.57] - Triangle Prism I>I

[0,0.785] - Square Pyramid II>

[0,2.355] - Tetrahedron I>>

[0,3.14] - Inverted I>I

Animated GIF : http://hddb.teamikaria.com/forum/viewtopic.php?p=23111#p23111

Here's probably the most complicated hypertoric donut ring in 7D : ((((II)I)I)((II)I))

Full write-up with many explore functions, and fantastic gifs of passing the shape through 3D, at 5 different angles:

http://hddb.teamikaria.com/forum/viewtopic.php?p=22704#p22704

Below this post are many more hypertoric shapes in 5D and 7D.

As usual, enjoy!


userPhilip

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Posted: 12/27/2014; 12:43:44 AM blueArrow

Here are 56 shapes that live in the 5th dimension. Starting with the hypertoric donut rings, we pass by the 4-sphere IOOOO, various n-spheric n-cones, the 5-simplex I>>>> , various triangular tapertopes, the 5-cube IIIII , and various cartesian product hyperprisms. Not including all possible toric shapes with a 2,3,4D crosscut, like a triangular toriglome I>(OOO), pyramidal tigroid II>[(O)(O)], or a pentachoronal torus I>>>(O), to name a few. Those strange shapes are the rolling surfaces of a +6D hypercylinder. Also included is the superior toratopic notation, i.e. (((II)I)(II)) , for hypertoric shapes.

IOOO(O) / ((II)III) - GLOMITORUS - (sqrt(x²+y²) -a)² +z²+w²+v² - b²

IOO[(O)(O)] / ((II)(II)I) - SPHERITIGER : (sqrt(x²+y²) -a)² + (sqrt(z²+w²) -b)² +v² - c²

IOO(OO) / ((III)II) - SPHERISPHERE : (sqrt(x²+y²+z²) -a)² +w²+v² - b²

IOO(O)(O) / (((II)I)II) -SPHERIDITORUS : (sqrt((sqrt(x²+y²) -a)² +z²) -b)² +w²+v² - c²

IO[(OO)(O)] / ((III)(II)) - CYLSPHERINTIGROID : (sqrt(x²+y²+z²) -a)² + (sqrt(w²+v²) -b)² - c²

IO[(O)(O)](O) / (((II)I)(II)) - TIGER TORUS : (sqrt((sqrt(x²+y²) -a)² +z²) -c)² + (sqrt(w²+v²) -b)² - d²

IO(OOO) / ((IIII)I) - TORIGLOME : (sqrt(x²+y²+z²+w²) -a)² +v² - b²

IO(OO)(O) / (((II)II)I) - TORISPHERITORUS : (sqrt((sqrt(x²+y²) -a)² +z²+w²) -b)² +v² - c²

IO(O)[(O)(O)] / (((II)(II))I) - TORATIGER (sqrt((sqrt(x²+y²) -a)² + (sqrt(z²+w²) -b)²) -c)² +v² - d²

IO(O)(OO) / (((III)I)I) - DITORISPHERE : (sqrt((sqrt(x²+y²+z²) -a)² +w²) -b)² +v² - c²

IO(O)(O)(O) / ((((II)I)I)I) - TRITORUS : (sqrt((sqrt((sqrt(x²+y²) -a)² +z²) -b)² +w²) -c)² +v² - d²

IOOOO / (IIIII) - PENTASPHERE : sqrt(x²+y²+z²+w²+v²) - a²

IOOO> - GLONE: |sqrt(x²+y²+z²+w²) - 2v| + sqrt(x²+y²+z²+w²) - a²

IOOOI - GLOMINDER: |sqrt(x²+y²+z²+w²) - 2v| + |sqrt(x²+y²+z²+w²) + 2v| - a²

IOO>> - DISPHONE: ||sqrt(x²+y²+z²) - 2w| + sqrt(x²+y²+z²) - 4v| + |sqrt(x²+y²+z²) - 2w| + sqrt(x²+y²+z²) - a²

IOO>I - SPHONINDER: ||sqrt(x²+y²+z²) - 2w| + sqrt(x²+y²+z²) - 2v| + ||sqrt(x²+y²+z²) - 2w| + sqrt(x²+y²+z²) + 2v| - a²

IOOIO - CYLSPHERINDER: |sqrt(x²+y²+z²) - sqrt(w²+v²)| + |sqrt(x²+y²+z²) + sqrt(w²+v²)| - a²

IOOI> - SPHERINDRONE: ||sqrt(x²+y²+z²) -w| + |sqrt(x²+y²+z²) +w| - 2v| + |sqrt(x²+y²+z²) -w| + |sqrt(x²+y²+z²) +w| - a

IOOII - CUBSPHERINDER: |sqrt(x²+y²+z²) - |w-v|-|w+v|| + |sqrt(x²+y²+z²) + |w-v|+|w+v|| - a²

IO>>> TRICONE: |||sqrt(x²+y²)-2z|+sqrt(x²+y²)-2w|+|sqrt(x²+y²)-2z|+sqrt(x²+y²) -4v| + ||sqrt(x²+y²)-2z|+sqrt(x²+y²)-2w|+|sqrt(x²+y²)-2z|+sqrt(x²+y²) -a²

IO>>I - DICONINDER: |||sqrt(x²+y²)-2z|+sqrt(x²+y²)-2w|+|sqrt(x²+y²)-2z|+sqrt(x²+y²)-4v| + |||sqrt(x²+y²)-2z|+sqrt(x²+y²)-2w|+|sqrt(x²+y²)-2z|+sqrt(x²+y²)-4v| -a²

IO>IO - CYLCONINDER: ||sqrt(x²+y²)-2z|+sqrt(x²+y²) - sqrt(w²+v²)| + ||sqrt(x²+y²)-2z|+sqrt(x²+y²) + sqrt(w²+v²)| - a

IO>I> - CONINDRONE: |||sqrt(x²+y²)-2z|+sqrt(x²+y²)-w|+||sqrt(x²+y²)-2z|+sqrt(x²+y²)+w|-4v| + ||sqrt(x²+y²)-2z|+sqrt(x²+y²)-w|+||sqrt(x²+y²)-2z|+sqrt(x²+y²)+w| - a

IO>II - CONE DIPRISM: ||sqrt(x²+y²)-2z|+sqrt(x²+y²) - |w-v|-|w+v|| + ||sqrt(x²+y²)-2z|+sqrt(x²+y²) + |w-v|+|w+v|| - a

IIOO> - DUOCYLINDRONE: ||sqrt(x²+y²)-sqrt(z²+w²)|+|sqrt(x²+y²)+sqrt(z²+w²)| -2v| + |sqrt(x²+y²)-sqrt(z²+w²)|+|sqrt(x²+y²)+sqrt(z²+w²)| - a²

IIO>> - DICYLINDRONE: |||sqrt(x²+y²)-z|+|sqrt(x²+y²)+z|-2w|+|sqrt(x²+y²)-z|+|sqrt(x²+y²)+z|-4v|+||sqrt(x²+y²)-z|+|sqrt(x²+y²)+z|-2w|+|sqrt(x²+y²)-z|+|sqrt(x²+y²)+z|-a

IIO>I - CYLINDRONINDER: |||sqrt(x²+y²)-z|+|sqrt(x²+y²)+z|-2w|+|sqrt(x²+y²)-z|+|sqrt(x²+y²)+z|-4v|+|||sqrt(x²+y²)-z|+|sqrt(x²+y²)+z|-2w|+|sqrt(x²+y²)-z|+|sqrt(x²+y²)+z|+4v|-a

I>>>> - HEXATERON: |||||x|-2y|+|x|-2z|+||x|-2y|+|x|-4w|+|||x|-2y|+|x|-2z|+||x|-2y|+|x| -8v| + ||||x|-2y|+|x|-2z|+||x|-2y|+|x|-4w|+|||x|-2y|+|x|-2z|+||x|-2y|+|x| - a

I>>>I - PENTACHORINDER: |||||x|-2y|+|x|-2z|+||x|-2y|+|x|-4w|+|||x|-2y|+|x|-2z|+||x|-2y|+|x|-8v|+|||||x|-2y|+|x|-2z|+||x|-2y|+|x|-4w|+|||x|-2y|+|x|-2z|+||x|-2y|+|x|+8v| - a

I>>IO - CYLTETRAHEDRINDER: ||||x|-2y|+|x|-2z| + ||x|-2y|+|x| - 2sqrt(w²+v²)| + ||||x|-2y|+|x|-2z| + ||x|-2y|+|x| + 2sqrt(w²+v²)| - a

I>>I> - TETRAHEDRINDER PYRAMID: |||||x|-2y|+|x|-2z|+||x|-2y|+|x|-2w|+||||x|-2y|+|x|-2z|+||x|-2y|+|x|+2w|-4v|+||||x|-2y|+|x|-2z|+||x|-2y|+|x|-2w|+||||x|-2y|+|x|-2z|+||x|-2y|+|x|+2w| - a

I>>II - TETRAHEDRON DIPRISM: ||||x|-2y|+|x|-2z| + ||x|-2y|+|x| - |w-v|-|w+v|| + ||||x|-2y|+|x|-2z| + ||x|-2y|+|x| + |w-v|+|w+v||

I>IO> - CYLTRIANGLINDRONE: ||||x|-2y|+|x|-sqrt(z²+w²)|+|||x|-2y|+|x|+sqrt(z²+w²)|-4v|+|||x|-2y|+|x|-sqrt(z²+w²)|+|||x|-2y|+|x|+sqrt(z²+w²)| - a

I>IOI - CUBTRIANGLINDER: ||sqrt(x²+y²)-z|+|sqrt(x²+y²)+z| - ||w|-2v|-|w|| + ||sqrt(x²+y²)-z|+|sqrt(x²+y²)+z| + ||w|-2v|+|w|| - a²

I>I>> - TRIANGLINDER DIPYRAMID: |||||x|-2y|+|x|-z|+|||x|-2y|+|x| +z|-2w|+|||x|-2y|+|x|-z|+|||x|-2y|+|x|+z|-4v|+||||x|-2y|+|x|-z|+|||x|-2y|+|x| +z|-2w|+|||x|-2y|+|x|-z|+|||x|-2y|+|x|+z| - a

I>I>I - TRIANGLINDER PYRAMINDER: |||||x|-2y|+|x|-z|+|||x|-2y|+|x| +z|-2w|+|||x|-2y|+|x|-z|+|||x|-2y|+|x|+z|-4v| + |||||x|-2y|+|x|-z|+|||x|-2y|+|x| +z|-2w|+|||x|-2y|+|x|-z|+|||x|-2y|+|x|+z|+4v| - a

I>II> - TRIANGLE DIPRISM PYRAMID: ||||x|-2y|+|x|-|z-w|-|z+w||+|||x|-2y|+|x|+|z-w|+|z+w|| -4v| + |||x|-2y|+|x|-|z-w|-|z+w||+|||x|-2y|+|x|+|z-w|+|z+w|| - a

I>III - TRIANGLE TRIPRISM: |||x-y|+|x+y|-2z|+||x-y|+|x+y|+2z| -||w|-2v|-|w|| + |||x-y|+|x+y|-2z|+||x-y|+|x+y|+2z| +||w|-2v|+|w|| - a

II>>> - SQUARE TRIPYRAMID: ||||x-y|+|x+y|-3z|+|x-y|+|x+y|-3w|+||x-y|+|x+y|-3z|+|x-y|+|x+y| -3v| + |||x-y|+|x+y|-3z|+|x-y|+|x+y|-3w|+||x-y|+|x+y|-3z|+|x-y|+|x+y| - a

II>>I - SQUARE DIPYRAMID PRISM: ||||x-y|+|x+y|-3z|+|x-y|+|x+y|-3w|+||x-y|+|x+y|-3z|+|x-y|+|x+y|-3v|+||||x-y|+|x+y|-3z|+|x-y|+|x+y|-3w|+||x-y|+|x+y|-3z|+|x-y|+|x+y|+3v| - a

II>IO - CYLHEMOCTAHEDRINDER : |||x-y|+|x+y|-2z| + |x-y|+|x+y| - 3sqrt(w²+v²)| + |||x-y|+|x+y|-2z| + |x-y|+|x+y| + 3sqrt(w²+v²)| - a²

II>I> - SQUARE PYRAMID PRISMID: ||||x-y|+|x+y|-3z|+|x-y|+|x+y|-2w|+|||x-y|+|x+y|-3z|+|x-y|+|x+y|+2w|-4v|+|||x-y|+|x+y|-3z|+|x-y|+|x+y|-2w|+|||x-y|+|x+y|-3z|+|x-y|+|x+y|+2w| - a

II>II - SQUARE PYRAMID DIPRISM: |||x-y|+|x+y|-2z| + |x-y|+|x+y| - |w-v|-|w+v|| + |||x-y|+|x+y|-2z| + |x-y|+|x+y| + |w-v|+|w+v|| - a²

IOIOI - DUOCYLDYINDER: ||sqrt(x²+y²)-sqrt(z²+w²)|+|sqrt(x²+y²)+sqrt(z²+w²)|-2v| + ||sqrt(x²+y²)-sqrt(z²+w²)|+|sqrt(x²+y²)+sqrt(z²+w²)|+2v| - a²

IIIO> - CUBINDRONE: |||x-y|+|x+y|-sqrt(z²+w²)|+||x-y|+|x+y|+sqrt(z²+w²)| -2v| + ||x-y|+|x+y|-sqrt(z²+w²)|+||x-y|+|x+y|+sqrt(z²+w²)| - a

III>> - CUBE DIPYRAMID: ||||x-y|+|x+y|-2z|+||x-y|+|x+y|+2z|-4w|+||x-y|+|x+y|-2z|+||x-y|+|x+y|+2z| -4v| + |||x-y|+|x+y|-2z|+||x-y|+|x+y|+2z|-4w|+||x-y|+|x+y|-2z|+||x-y|+|x+y|+2z| - a

III>I - CUBE PYRAMID PRISM: ||||x-y|+|x+y|-2z|+||x-y|+|x+y|+2z|-4w|+||x-y|+|x+y|-2z|+||x-y|+|x+y|+2z| -4v| + ||||x-y|+|x+y|-2z|+||x-y|+|x+y|+2z|-4w|+||x-y|+|x+y|-2z|+||x-y|+|x+y|+2z| +4v| - a

IIIIO - TESSERINDER: |||x-y|+|x+y|-2z| + ||x-y|+|x+y|+2z| -sqrt(w²+v²)| + |||x-y|+|x+y|-2z| + ||x-y|+|x+y|+2z| +sqrt(w²+v²)| - a

IIII> - TESSERACT PYRAMID: |||x-y|+|x+y|-|z-w|-|z+w|| + ||x-y|+|x+y|+|z-w|+|z+w|| -2v| + ||x-y|+|x+y|-|z-w|-|z+w|| + ||x-y|+|x+y|+|z-w|+|z+w|| - a

IIIII - PENTERACT: |||x-y|+|x+y|-|z-w|-|z+w|| + ||x-y|+|x+y|+|z-w|+|z+w|| -2v| + |||x-y|+|x+y|-|z-w|-|z+w|| + ||x-y|+|x+y|+|z-w|+|z+w|| +2v| - a

IOO[I>] - SPHENTRIANGLINDER: |sqrt(x²+y²+z²) - ||w|-2v|-|w|| + |sqrt(x²+y²+z²) + ||w|-2v|+|w|| - a

IO>[I>] - CONTRIANGLINDER : ||sqrt(x²+y²)-2z|+sqrt(x²+y²) - ||w|-2v|-|w|| + ||sqrt(x²+y²)-2z|+sqrt(x²+y²) + ||w|-2v|+|w|| - a²

II>[I>] - HEMOCTAHEDROTRIANGLINDER: |||x-y|+|x+y|-2z| + |x-y|+|x+y| - 2||w|-2v|-|w|| + |||x-y|+|x+y|-2z| + |x-y|+|x+y| + 2||w|-2v|+|w|| - a²

I>>[I>] - TETRAHEDROTRIANGLINDER: ||||x|-2y|+|x|-2z| + ||x|-2y|+|x| - 2||w|-2v|-|w|| + ||||x|-2y|+|x|-2z| + ||x|-2y|+|x| + 2||w|-2v|+|w|| - a

I>[I>]I - DUOTRIANGLINDER PRISM: ||||x|-2y|+|x|-||z|-2w|-|z||+|||x|-2y|+|x|+||z|-2w|+|z|| -4v| + ||||x|-2y|+|x|-||z|-2w|-|z||+|||x|-2y|+|x|+||z|-2w|+|z|| +4v| - a

I>[I>]> - DUOTRIANGLINDRIC PYRAMID: ||||x|-2y|+|x|-||z|-2w|-|z||+|||x|-2y|+|x|+||z|-2w|+|z|| -4v| + |||x|-2y|+|x|-||z|-2w|-|z||+|||x|-2y|+|x|+||z|-2w|+|z|| - a


userPhilip

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Posted: 1/23/2015; 9:34:22 PM blueArrow

Here's some neat shapes in 6D :

IO[IO]>I : 6D Duocylindroninder , prism of the pyramid of the 4D duocylinder. Has infinite 5D duocylindrical pyramids XYZWV stacked within line segment U

XYZWVU

I - 1D Line : Start with line in 1-plane X

|x| = a

IO - 2D Circle : bisecting rotate LINE around stationary point, along X into Y

√(x²+y²) = a

IOI - 3D Cylinder : extend CIRCLE along line into Z

|√(x²+y²) - z| + |√(x²+y²) + z| = a

IOIO = IO[IO] - 4D Duocylinder : bisecting rotate CYLINDER around stationary plane XY, along Z into W

|√(x²+y²) - √(z²+w²)| + |√(x²+y²) + √(z²+w²)| = a

IO[IO]> - 5D Duocylindrone : shrink DUOCYLINDER to point while extending along V

||√(x²+y²)-√(z²+w²)| + |√(x²+y²)+√(z²+w²)| + 2v| + |√(x²+y²)-√(z²+w²)| + |√(x²+y²)+√(z²+w²)| = a

IO[IO]>I - 6D Duocylindroninder : extend DUOCYLINDRONE along line into U

|||√(x²+y²)-√(z²+w²)|+|√(x²+y²)+√(z²+w²)|+2v| + |√(x²+y²)-√(z²+w²)|+|√(x²+y²)+√(z²+w²)| - 4u| + |||√(x²+y²)-√(z²+w²)|+|√(x²+y²)+√(z²+w²)|+2v| + |√(x²+y²)-√(z²+w²)|+|√(x²+y²)+√(z²+w²)| + 4u| = a

———————

IO>I[IO] = IO>[IOI] : 6D Cubconinder , cartesian product of cone times cylinder. It has infinite cones XYZ stacked within a cylinder-segment WVU.

XYZWVU

I - 1D Line : Start with line in 1-plane X

|x| = a

IO - 2D Circle : bisecting rotate LINE around stationary point, along X into Y

√(x²+y²) = a

IO> - 3D Cone : shrink CIRCLE to point while extending along Z

|√(x²+y²) + 2z| + √(x²+y²) = a

IO>I - 4D Coninder : extend CONE along line into W

||√(x²+y²)+2z| + √(x²+y²) - 2w| + ||√(x²+y²)+2z| + √(x²+y²) + 2w| = a

IO>IO = IO>[IO] - 5D Cylconinder : bisecting rotate CONINDER around stationary plane XYZ, along W into V

||√(x²+y²)+2z| + √(x²+y²) - 2√(w²+v²)| + ||√(x²+y²)+2z| + √(x²+y²) + 2√(w²+v²)| = a

IO>IOI = IO>[IOI] - Cubconinder : extend CYLCONINDER along line into U

||√(x²+y²)+2z| + √(x²+y²) - |√(w²+v²)-u| - |√(w²+v²)+u|| + ||√(x²+y²)+2z| + √(x²+y²) + |√(w²+v²)-u| + |√(w²+v²)+u|| = a

-----------


Parsing the 3D midsections and combining with a triple rotate function:

IOO>II : 6D Sphone Diprism , cubic prism of the 4D spherical cone. Has infinite 4D sphones XYZW stacked within a square-segment VU

XYZWVU

I - 1D Line : Start with line in 1-plane X

|x| = a

IO - 2D Circle : bisecting rotate LINE around stationary point, along X into Y

√(x²+y²) = a

IOO - 3D Sphere : bisecting rotate CIRCLE around stationary axis X, along Y into Z

√(x²+y²+z²) = a

IOO> - 4D Sphone : shrink SPHERE to point while extending along W

|√(x²+y²+z²) + 2w| + √(x²+y²+z²) = a

IOO>I - 5D Sphoninder : extend SPHONE along line into V

||√(x²+y²+z²)+2w| + √(x²+y²+z²) - 2v| + ||√(x²+y²+z²)+2w| + √(x²+y²+z²) + 2v| = a

IOO>II - 6D Sphone Diprism : extend SPHONINDER along line into U

||√(x²+y²+z²)+2w| + √(x²+y²+z²) - |v-u| - |v+u|| + ||√(x²+y²+z²)+2w| + √(x²+y²+z²) + |v-u| + |v+u|| = a

abs(abs(sqrt(x^2+y^2+z^2) + 2w) + sqrt(x^2+y^2+z^2) - abs(v-u) - abs(v+u)) + abs(abs(sqrt(x^2+y^2+z^2) + 2w) + sqrt(x^2+y^2+z^2) + abs(v-u) + abs(v+u)) = a

3D midsections

—————————————

XYZWVU

IiiiII , iOiiII , iiOiII - cube III , YZW cut . Also identical to iii>II as the other cube-section along XYZ cut

abs(abs(sqrt(x^2+0^2+0^2) - 2*0) + sqrt(x^2+0^2+0^2) - abs(y-z) - abs(y+z)) + abs(abs(sqrt(x^2+0^2+0^2) - 2*0) + sqrt(x^2+0^2+0^2) + abs(y-z) + abs(y+z)) = a

Iii>iI - triangle prism I>I , YZV cut

abs(abs(sqrt(x^2+0^2+0^2) - 2y) + sqrt(x^2+0^2+0^2) - abs(0-z) - abs(0+z)) + abs(abs(sqrt(x^2+0^2+0^2) - 2y) + sqrt(x^2+0^2+0^2) + abs(0-z) + abs(0+z)) = a

IOi>ii - cone IO> , ZVU cut

abs(abs(sqrt(x^2+y^2+0^2) - 2z) + sqrt(x^2+y^2+0^2) - abs(0-0) - abs(0+0)) + abs(abs(sqrt(x^2+y^2+0^2) - 2z) + sqrt(x^2+y^2+0^2) + abs(0-0) + abs(0+0)) = a

IOiiiI - cylinder IOI , ZWV

abs(abs(sqrt(x^2+y^2+0^2) - 2*0) + sqrt(x^2+y^2+0^2) - abs(0-z) - abs(0+z)) + abs(abs(sqrt(x^2+y^2+0^2) - 2*0) + sqrt(x^2+y^2+0^2) + abs(0-z) + abs(0+z)) = a

IOOiii - sphere , WVU cut

abs(abs(sqrt(x^2+y^2+z^2) - 2*0) + sqrt(x^2+y^2+z^2) - abs(0-0) - abs(0+0)) + abs(abs(sqrt(x^2+y^2+z^2) - 2*0) + sqrt(x^2+y^2+z^2) + abs(0-0) + abs(0+0)) = a

IOOiii to iii>II rotate function

abs(abs(sqrt((x*sin(a))^2+(y* sin(b))^2+(z*sin(c))^2) - 2(x*cos(a))) + sqrt((x*sin(a))^2+(y* sin(b))^2+(z*sin(c))^2) - abs((y*cos(b))-(z*cos(c))) - abs((y*cos(b))+(z*cos(c)))) + abs(abs(sqrt((x*sin(a))^2+(y* sin(b))^2+(z*sin(c))^2) - 2(x*cos(a))) + sqrt((x*sin(a))^2+(y* sin(b))^2+(z*sin(c))^2) + abs((y*cos(b))-(z*cos(c))) + abs((y*cos(b))+(z*cos(c)))) = d

Set range to 0 < a,b,c < 1.57 for 90 deg turn in three independent circles of rotation, O < d < 15 for size scaling. Set XYZbox to -8, +8.

[a,b,c]-midsection positions

[1.57,1.57,1.57] - sphere IOO

[0,1.57,1.57] - cone IO>

[1.57,0,1.57] - cylinder IOI

[1.57,1.57,0] - cylinder IOI

[1.57,0,0] - cube III

[0,1.57,0] - triangle prism I>I

[0,0,1.57] - triangle prism I>I

[0,0,0] - cube III


userPhilip

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Posted: 2/6/2015; 5:34:33 PM blueArrow

Cool new 5D shape I checked out recently:

IO>I> : 5D Conindrone , 5D pyramid of the 4D prism of the 3D cone

* - Point : Starting element

I - Line : extend Point along X

|x| = a

IO - Circle : bisecting rotate Line around point, along X into Y

√(x² + y²) = a

IO> - Cone : taper circle to point while extending along Z

|√(x²+y²) + 2z| + √(x²+y²) = a

IO>I - Coninder : extend Cone along W

||√(x²+y²)+2z|+√(x²+y²) - 2w| + ||√(x²+y²)+2z|+√(x²+y²) + 2w| = a

IO>I> - Conindrone : taper Coninder to point while extending along V

|||√(x²+y²)+2z|+√(x²+y²)-2w| + ||√(x²+y²)+2z|+√(x²+y²)+2w| + 4v| + ||√(x²+y²)+2z|+√(x²+y²)-2w| + ||√(x²+y²)+2z|+√(x²+y²)+2w| = a

abs(abs(abs(sqrt(x^2+y^2)+2z)+sqrt(x^2+y^2)-2w) + abs(abs(sqrt(x^2+y^2)+2z)+sqrt(x^2+y^2)+2w) + 4v) + abs(abs(sqrt(x^2+y^2)+2z)+sqrt(x^2+y^2)-2w) + abs(abs(sqrt(x^2+y^2)+2z)+sqrt(x^2+y^2)+2w) = a

---


3D Midsections of IO>I> along XYZWV

Circumradius = d = 10

XYZbox = -6 / +6

Slide 'a' and 'b' to move in to 4D and 5D

• ii>I> - square pyramid II> , XY cut

abs(abs(abs(sqrt(a^2+b^2)+2x)+sqrt(a^2+b^2)-2y) + abs(abs(sqrt(a^2+b^2)+2x)+sqrt(a^2+b^2)+2y) + 4z) + abs(abs(sqrt(a^2+b^2)+2x)+sqrt(a^2+b^2)-2y) + abs(abs(sqrt(a^2+b^2)+2x)+sqrt(a^2+b^2)+2y) = d

• IiiI> - square pyramid II> , YZ cut

abs(abs(abs(sqrt(x^2+a^2)+2b)+sqrt(x^2+a^2)-2y) + abs(abs(sqrt(x^2+a^2)+2b)+sqrt(x^2+a^2)+2y) + 4z) + abs(abs(sqrt(x^2+a^2)+2b)+sqrt(x^2+a^2)-2y) + abs(abs(sqrt(x^2+a^2)+2b)+sqrt(x^2+a^2)+2y) = d

• Ii>i> - tetrahedron I>> , YW cut

abs(abs(abs(sqrt(x^2+a^2)+2z)+sqrt(x^2+a^2)-2b) + abs(abs(sqrt(x^2+a^2)+2z)+sqrt(x^2+a^2)+2b) + 4y) + abs(abs(sqrt(x^2+a^2)+2z)+sqrt(x^2+a^2)-2b) + abs(abs(sqrt(x^2+a^2)+2z)+sqrt(x^2+a^2)+2b) = d

• Ii>Ii - triangle prism I>I , YV cut

abs(abs(abs(sqrt(x^2+a^2)+2z)+sqrt(x^2+a^2)-2y) + abs(abs(sqrt(x^2+a^2)+2z)+sqrt(x^2+a^2)+2y) + 4b) + abs(abs(sqrt(x^2+a^2)+2z)+sqrt(x^2+a^2)-2y) + abs(abs(sqrt(x^2+a^2)+2z)+sqrt(x^2+a^2)+2y) = d

• IOii> - cone IO> , ZW cut

abs(abs(abs(sqrt(x^2+y^2)+2a)+sqrt(x^2+y^2)-2b) + abs(abs(sqrt(x^2+y^2)+2a)+sqrt(x^2+y^2)+2b) + 4z) + abs(abs(sqrt(x^2+y^2)+2a)+sqrt(x^2+y^2)-2b) + abs(abs(sqrt(x^2+y^2)+2a)+sqrt(x^2+y^2)+2b) = d

• IOiIi - cylinder IOI , ZV cut

abs(abs(abs(sqrt(x^2+y^2)+2a)+sqrt(x^2+y^2)-2z) + abs(abs(sqrt(x^2+y^2)+2a)+sqrt(x^2+y^2)+2z) + 4b) + abs(abs(sqrt(x^2+y^2)+2a)+sqrt(x^2+y^2)-2z) + abs(abs(sqrt(x^2+y^2)+2a)+sqrt(x^2+y^2)+2z) = d

---------


• IiiI> to IO>ii dual rotate function, y -> a , z -> b

abs(abs(abs(sqrt(x^2+(y*cos(a))^2)+2(z*cos(b)))+sqrt(x^2+(y*cos(a))^2)-2(y*sin(a))) + abs(abs(sqrt(x^2+(y*cos(a))^2)+2(z*cos(b)))+sqrt(x^2+(y*cos(a))^2)+2(y*sin(a))) + 4(z*sin(b))) + abs(abs(sqrt(x^2+(y*cos(a))^2)+2(z*cos(b)))+sqrt(x^2+(y*cos(a))^2)-2(y*sin(a))) + abs(abs(sqrt(x^2+(y*cos(a))^2)+2(z*cos(b)))+sqrt(x^2+(y*cos(a))^2)+2(y*sin(a))) = 10

• Ii>i> to IOiIi dual rotate, y -> a , z -> b

abs(abs(abs(sqrt(x^2+(y*cos(a))^2)+2(z*sin(b)))+sqrt(x^2+(y*cos(a))^2)-2(z*cos(b))) + abs(abs(sqrt(x^2+(y*cos(a))^2)+2(z*sin(b)))+sqrt(x^2+(y*cos(a))^2)+2(z*cos(b))) + 4(y*sin(a))) + abs(abs(sqrt(x^2+(y*cos(a))^2)+2(z*sin(b)))+sqrt(x^2+(y*cos(a))^2)-2(z*cos(b))) + abs(abs(sqrt(x^2+(y*cos(a))^2)+2(z*sin(b)))+sqrt(x^2+(y*cos(a))^2)+2(z*cos(b))) = 10

— better rotate morphs

[a,b] - 3D Midsection Positions

[0,0] - cylinder IOI

[1.57,0] - square pyramid II>

[0,1.57] - cone IO>

[1.57,1.57] - tetrahedron I>>

-----


• Ii>i> to IOiIi : y -> a , z -> b, Dual Rotate + Translate for full control over all angles and depth in 4D and 5D

abs(abs(abs(sqrt(x^2+(y*cos(b) - a*sin(b))^2)+2(z*sin(d) + c*cos(d)))+sqrt(x^2+(y*cos(b) - a*sin(b))^2)-2(z*cos(d) - c*sin(d))) + abs(abs(sqrt(x^2+(y*cos(b) - a*sin(b))^2)+2(z*sin(d) + c*cos(d)))+sqrt(x^2+(y*cos(b) - a*sin(b))^2)+2(z*cos(d) - c*sin(d))) + 4(y*sin(b) + a*cos(b))) + abs(abs(sqrt(x^2+(y*cos(b) - a*sin(b))^2)+2(z*sin(d) + c*cos(d)))+sqrt(x^2+(y*cos(b) - a*sin(b))^2)-2(z*cos(d) - c*sin(d))) + abs(abs(sqrt(x^2+(y*cos(b) - a*sin(b))^2)+2(z*sin(d) + c*cos(d)))+sqrt(x^2+(y*cos(b) - a*sin(b))^2)+2(z*cos(d) - c*sin(d))) = 10

-6 < a,c < +6

0 < b,d < 1.57

‘a’ slides, ‘b’ rotates in 4D

‘c’ slides, ‘d’ rotates in 5D

[b,d] - 3D Midsection Positions

[0,0] - cylinder IOI

[1.57,0] - square pyramid II>

[0,1.57] - cone IO>

[1.57,1.57] - tetrahedron I>>


userPhilip

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Posted: 2/19/2015; 2:02:26 PM blueArrow

I made some more CalcPlot3D scripts, and added to the Hypertorus Explore Function Library:

http://hddb.teamikaria.com/forum/viewtopic.php?p=23287#p23287

And the animated GIF for Conindrone IO>I> :

http://hddb.teamikaria.com/forum/viewtopic.php?p=23507#p23507

Also, quite recently, I put together a nice GIF gallery to introduce 4D hypertoric shapes:

http://hddb.teamikaria.com/forum/viewtopic.php?p=23519#p23519

I came across a neat 2D graphing program as well, and decided to create 3D shapes in it, with full slide and rotate control in 3D. What I like about this calculator app, is how you can save the function and custom settings as an html. So, all one has to do is click the link, and it'll take you right to the script in the app. This little feature can really improve the accessibility of calcplot3d for the layman, who might not be very well versed in tuning calcplot, setting slider ranges, graphing functions, etc.

Desmos Scripts for 2D Exploration of 3D/4D Shapes:

For 3D shapes, adjust 'a' for sliding and 'b' for rotating. Some 3D shapes use two rotation parameters, in which case 'b' and 'c' .

• Exploring 3D Torus in 2D, m-slide / n-rotate

https://www.desmos.com/calculator/otpsrtlnww

• Exploring 3D Cylinder in 2D, m-slide / n-rotate

https://www.desmos.com/calculator/otpeykbx8g

• Exploring 3D Cube in 2D

https://www.desmos.com/calculator/fgqzkxuntu

• Exploring 3D Cone in 2D

https://www.desmos.com/calculator/w3xptfnyhb

• Exploring 3D Tetrahedron in 2D

https://www.desmos.com/calculator/ymijhsdgxc

• Exploring 3D Triangle Prism in 2D

https://www.desmos.com/calculator/1mmbj9339n

• Exploring 3D Square Pyramid in 2D

https://www.desmos.com/calculator/xupqv2qyq7

These two are 4D, and use a dual translate + rotate function.

'a' slides, 'b' rotates in 3D

'c' slides, 'd' rotates in 4D

• Exploring 4D Tiger in 2D

https://www.desmos.com/calculator/tidajl5xxg

• Exploring 4D Ditorus in 2D

https://www.desmos.com/calculator/emdbucmoqg


userPhilip

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Posted: 3/15/2015; 6:20:53 PM blueArrow

I recently updated my Explore Function Libraries, for hypertoric and new hyperprismic shapes:

http://hi.gher.space/forum/viewtopic.php?p=23578#p23578


userPhilip

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Posted: 4/22/2015; 6:35:28 PM blueArrow

Here's another wild creation of mine: a single mathematical function that can define 7 uniquely different 3D shapes. It's actually a surface equation for a particular 7D shape. Solving for 3D plane equations, by setting a 4-plane to zero, will result in different shapes.

How I derived the " Super-Function " :

IOO>II> - 7D Cubsphonindrone, pyramid of (sphone,square) prism

• I - 1D Line : Start with line in 1-plane X

|x| = a

• IO - 2D Circle : bisecting rotate Line around origin, along X into Y

√(x²+y²) = a

• IOO - 3D Sphere : bisecting rotate Circle around X, along Y into Z

√(x²+y²+z²) = a

• IOO> - 4D Sphone : scale Sphere to a point along W

|√(x²+y²+z²) + 2w| + √(x²+y²+z²) = a

• IOO>I - 5D Sphoninder : extend Sphone along V, sphone-xyzw times line-v

||√(x²+y²+z²)+2w|+√(x²+y²+z²) - 2v| + ||√(x²+y²+z²)+2w|+√(x²+y²+z²) + 2v| = a

• IOO>II - 6D Cubsphoninder : extend Sphoninder along U, sphone-xyzw times square-vu

||√(x²+y²+z²)+2w|+√(x²+y²+z²) - |v-u|-|v+u|| + ||√(x²+y²+z²)+2w|+√(x²+y²+z²) + |v-u|+|v+u|| = a

• IOO>II> - 7D Cubsphonindrone : scale Cubsphoninder to a point along T

|||√(x²+y²+z²)+2w|+√(x²+y²+z²)-|v-u|-|v+u||+||√(x²+y²+z²)+2w|+√(x²+y²+z²)+|v-u|+|v+u|| + 4t| + ||√(x²+y²+z²)+2w|+√(x²+y²+z²)-|v-u|-|v+u||+||√(x²+y²+z²)+2w|+√(x²+y²+z²)+|v-u|+|v+u|| = a

Full 7D Implicit Surface Equation:

f(x,y,z,w,v,u,t) = |||√(x²+y²+z²)+2w|+√(x²+y²+z²)-|v-u|-|v+u||+||√(x²+y²+z²)+2w|+√(x²+y²+z²)+|v-u|+|v+u|| + 4t| + ||√(x²+y²+z²)+2w|+√(x²+y²+z²)-|v-u|-|v+u||+||√(x²+y²+z²)+2w|+√(x²+y²+z²)+|v-u|+|v+u|| - a

-


3D Hyperplane Intersections of IOO>II> along XYZWVUT, i = dimension set to zero

• IOOiiii - IOO sphere, WVUT=0

abs(abs(abs(sqrt(x^2+y^2+z^2) +2*0)+sqrt(x^2+y^2+z^2) -abs(0-0)-abs(0+0)) + abs(abs(sqrt(x^2+y^2+z^2) +2*0)+sqrt(x^2+y^2+z^2) +abs(0-0)+abs(0+0)) + 4*0) + abs(abs(sqrt(x^2+y^2+z^2) +2*0)+sqrt(x^2+y^2+z^2) -abs(0-0)-abs(0+0)) + abs(abs(sqrt(x^2+y^2+z^2) +2*0)+sqrt(x^2+y^2+z^2) +abs(0-0)+abs(0+0)) = 20

• IOi>iii - IO> cone, ZVUT=0

abs(abs(abs(sqrt(x^2+y^2+0^2) +2z)+sqrt(x^2+y^2+0^2) -abs(0-0)-abs(0+0)) + abs(abs(sqrt(x^2+y^2+0^2) +2z)+sqrt(x^2+y^2+0^2) +abs(0-0)+abs(0+0)) + 4*0) + abs(abs(sqrt(x^2+y^2+0^2) +2z)+sqrt(x^2+y^2+0^2) -abs(0-0)-abs(0+0)) + abs(abs(sqrt(x^2+y^2+0^2) +2z)+sqrt(x^2+y^2+0^2) +abs(0-0)+abs(0+0)) = 20

• IOiiIii - IOI cylinder, ZWUT=0

abs(abs(abs(sqrt(x^2+y^2+0^2) +2*0)+sqrt(x^2+y^2+0^2) -abs(z-0)-abs(z+0)) + abs(abs(sqrt(x^2+y^2+0^2) +2*0)+sqrt(x^2+y^2+0^2) +abs(z-0)+abs(z+0)) + 4*0) + abs(abs(sqrt(x^2+y^2+0^2) +2*0)+sqrt(x^2+y^2+0^2) -abs(z-0)-abs(z+0)) + abs(abs(sqrt(x^2+y^2+0^2) +2*0)+sqrt(x^2+y^2+0^2) +abs(z-0)+abs(z+0)) = 20

• IiiiIIi - III cube, YZWT=0

abs(abs(abs(sqrt(x^2+0^2+0^2) +2*0)+sqrt(x^2+0^2+0^2) -abs(y-z)-abs(y+z)) + abs(abs(sqrt(x^2+0^2+0^2) +2*0)+sqrt(x^2+0^2+0^2) +abs(y-z)+abs(y+z)) + 4*0) + abs(abs(sqrt(x^2+0^2+0^2) +2*0)+sqrt(x^2+0^2+0^2) -abs(y-z)-abs(y+z)) + abs(abs(sqrt(x^2+0^2+0^2) +2*0)+sqrt(x^2+0^2+0^2) +abs(y-z)+abs(y+z)) = 20

• IiiiiI> - II> sq pyramid, YZWV=0

abs(abs(abs(sqrt(x^2+0^2+0^2) +2*0)+sqrt(x^2+0^2+0^2) -abs(0-y)-abs(0+y)) + abs(abs(sqrt(x^2+0^2+0^2) +2*0)+sqrt(x^2+0^2+0^2) +abs(0-y)+abs(0+y)) + 4z) + abs(abs(sqrt(x^2+0^2+0^2) +2*0)+sqrt(x^2+0^2+0^2) -abs(0-y)-abs(0+y)) + abs(abs(sqrt(x^2+0^2+0^2) +2*0)+sqrt(x^2+0^2+0^2) +abs(0-y)+abs(0+y)) = 20

• Iii>Iii - I>I triangle prism, YZUT=0

abs(abs(abs(sqrt(x^2+0^2+0^2) +2y)+sqrt(x^2+0^2+0^2) -abs(z-0)-abs(z+0)) + abs(abs(sqrt(x^2+0^2+0^2) +2y)+sqrt(x^2+0^2+0^2) +abs(z-0)+abs(z+0)) + 4*0) + abs(abs(sqrt(x^2+0^2+0^2) +2y)+sqrt(x^2+0^2+0^2) -abs(z-0)-abs(z+0)) + abs(abs(sqrt(x^2+0^2+0^2) +2y)+sqrt(x^2+0^2+0^2) +abs(z-0)+abs(z+0)) = 20

• Iii>ii> - I>> tetrahedron, YZVU=0

abs(abs(abs(sqrt(x^2+0^2+0^2) +2y)+sqrt(x^2+0^2+0^2) -abs(0-0)-abs(0+0)) + abs(abs(sqrt(x^2+0^2+0^2) +2y)+sqrt(x^2+0^2+0^2) +abs(0-0)+abs(0+0)) + 4z) + abs(abs(sqrt(x^2+0^2+0^2) +2y)+sqrt(x^2+0^2+0^2) -abs(0-0)-abs(0+0)) + abs(abs(sqrt(x^2+0^2+0^2) +2y)+sqrt(x^2+0^2+0^2) +abs(0-0)+abs(0+0)) = 20

--


Building the rotate function for turning the 3-plane to all 7 distinct intersections in 3D:

Take full 7D equation,

abs(abs(abs(sqrt(x^2+y^2+z^2) +2w)+sqrt(x^2+y^2+z^2) -abs(v-u)-abs(v+u)) + abs(abs(sqrt(x^2+y^2+z^2) +2w)+sqrt(x^2+y^2+z^2) +abs(v-u)+abs(v+u)) + 4t) + abs(abs(sqrt(x^2+y^2+z^2) +2w)+sqrt(x^2+y^2+z^2) -abs(v-u)-abs(v+u)) + abs(abs(sqrt(x^2+y^2+z^2) +2w)+sqrt(x^2+y^2+z^2) +abs(v-u)+abs(v+u)) = 20

establish rotate parameters,

Y -> W

Z -> V

(Y at W) -> U

(Z at V) -> T

which becomes,

y = (y*sin(a))

z = (z*sin(b))

w = ((y*cos(a))*sin(c))

v = ((z*cos(b))*sin(d))

u = ((y*cos(a))*cos(c))

t = ((z*cos(b))*cos(d))

Resulting in the final Explore Function (largest equation I ever wrote!):

abs(abs(abs(sqrt(x^2+(y*sin(a))^2+(z*sin(b))^2) +2((y*cos(a))*sin(c)))+sqrt(x^2+(y*sin(a))^2+(z*sin(b))^2) -abs(((z*cos(b))*sin(d))-((y*cos(a))*cos(c)))-abs(((z*cos(b))*sin(d))+((y*cos(a))*cos(c)))) + abs(abs(sqrt(x^2+(y*sin(a))^2+(z*sin(b))^2) +2((y*cos(a))*sin(c)))+sqrt(x^2+(y*sin(a))^2+(z*sin(b))^2) +abs(((z*cos(b))*sin(d))-((y*cos(a))*cos(c)))+abs(((z*cos(b))*sin(d))+((y*cos(a))*cos(c)))) + 4((z*cos(b))*cos(d))) + abs(abs(sqrt(x^2+(y*sin(a))^2+(z*sin(b))^2) +2((y*cos(a))*sin(c)))+sqrt(x^2+(y*sin(a))^2+(z*sin(b))^2) -abs(((z*cos(b))*sin(d))-((y*cos(a))*cos(c)))-abs(((z*cos(b))*sin(d))+((y*cos(a))*cos(c)))) + abs(abs(sqrt(x^2+(y*sin(a))^2+(z*sin(b))^2) +2((y*cos(a))*sin(c)))+sqrt(x^2+(y*sin(a))^2+(z*sin(b))^2) +abs(((z*cos(b))*sin(d))-((y*cos(a))*cos(c)))+abs(((z*cos(b))*sin(d))+((y*cos(a))*cos(c)))) = 20

Ranges:

0 < a,b,c,d < π/2

XYZbox = -10,10

Plot in 37 Cubes Resolution

-


• [a,b,c,d] - Adjustable Parameter Values for 3D Midsection Positions

IOO Sphere = [π/2,π/2,π/2,π/2], [π/2,π/2,0,π/2], [π/2,π/2,0,0]

IOI Cylinder = [π/2,0,π/2,π/2], [π/2,0,0,π/2], [0,π/2,0,π/2], [0,π/2,0,0]

IO> Cone = [0,π/2,π/2,π/2], [π/2,0,0,0], [0,π/2,π/2,0], [π/2,0,π/2,0]

III Cube = [0,0,0,π/2]

II> Square Pyramid = [0,0,0,0]

I>I Triangle Prism = [0,0,π/2,π/2]

I>> Tetrahedron = [0,0,π/2,0]

When a=1.57 or b=1.57, adjusting c or d, respectively, will have no effect. Only when they are equal to zero, will c and d rotate to another 3-plane.

And, of course, a CalcPlot3D Exploration Script:

http://hi.gher.space/forum/download/file.php?id=239


userPhilip

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Posted: 5/5/2015; 3:25:21 PM blueArrow

Found an interesting function for rotating a torus around a general hyperplane into 4D.

Take the equation for a torus,

(sqrt(x^2+y^2)-a)^2 + z^2 = b^2

Sweeping around plane xy will generate a 3-torus:

(sqrt((sqrt(x^2+y^2)-a)^2 +z^2)-b)^2 + w^2 = c^2

Sweeping around plane xz will generate a tiger, which is an exotic 3-torus, homeomorphic to T^3:

(sqrt(x^2+y^2)-a)^2 + (sqrt(z^2+w^2)-b)^2 = c^2

Sweeping around a general hyperplane, in between coordinate planes xy and xz will generate a 4D torus even stranger than T^3 or tiger:

(sqrt((x*sin(a)+(sqrt(z^2+w^2)-b)*cos(d))^2+y^2)-a)^2+(x*cos(a)-(sqrt(z^2+w^2)-b)*sin(d))^2 = c^2

Where a,b,c are diameters, and parameter d will change the plane of rotation between a T^3 and tiger. This will lead to two mirror image tilted tori. Adjusting d will dynamically change from xy to xz, to all angles in between.

--------


For CalcPlot, this function will make the 3D cross section of the above:

(sqrt((x*sin(a)+(sqrt(z^2+b^2)-3)*cos(a))^2+y^2)-3)^2+(x*cos(a)-(sqrt(z^2+b^2)-3)*sin(a))^2 = 1

set:

0 < a < 1.57

0 < b < 10

XYZbox = -10 , 10

'a' rotates the plane of rotation, 'b' slides along 4D, to merge the tori


userPhilip

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Posted: 5/5/2015; 5:15:52 PM blueArrow

Here's a really wild 5D equivalent. It makes several rotations and amazing morphs between ((((II)I)I)I) , (((II)(II))I) , and (((II)I)(II)). I recently learned that the two shapes, (((II)(II))I) and (((II)I)(II)) are exotic 4-tori in R^5, that are homeomorphic to the standard 4-torus ((((II)I)I)I). You get all three out of this function, plus the ability to rotate the specific shape a few ways. This is the wildest stuff I've seen yet with toratope functions:

(sqrt(((x*sin(d))*sin(a) + (sqrt((y*cos(b)-(sqrt((z*sin(c))^2+(x*cos(d))^2)-8)*sin(b))^2+(z*cos(c))^2)-4)*cos(a))^2 + (y*sin(b)+(sqrt((z*sin(c))^2+(x*cos(d))^2)-8)*cos(b))^2)-2.5)^2 + ((x*sin(d))*cos(a) - (sqrt((y*cos(b)-(sqrt((z*sin(c))^2+(x*cos(d))^2)-8)*sin(b))^2+(z*cos(c))^2)-4)*sin(a))^2 = 1

0 < a,b,c,d < 1.57

XYZbox = -17 , 17

a,b changes general plane of rotation

c,d rotates respective toratope


userPhilip

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Posted: 5/8/2015; 9:17:51 PM blueArrow

I played around with some coordinate systems and hypertorus equations. There are some very strange and amazing surfaces in this equation. And, some wild topology changes, too.

At its heart, the equation is basically a 4D hypertorus known as the tiger, desribed in spherindrical (sphere x line) coordinates. It also includes general plane of rotation adjustment, and translate + rotate in 4D:

1. Start with torus with major diameter parallel to xz,

(sqrt(x^2+z^2)-3)^2 + y^2 = 1

2. Set up rotation from x->y, adjusting 'c' will spin the torus around z,

x = (x*sin(c)+y*cos(c))

y = (x*cos(c)-y*sin(c))

(sqrt((x*sin(c)+y*cos(c))^2+z^2)-3)^2+(x*cos(c)-y*sin(c))^2 = 1

3. Non-intersecting rotate torus around plane yw into 4D. Adjusting 'c' will change general plane of rotation of the torus, between ditorus and tiger intercepts,

y = (sqrt(y^2+w^2)-5)

(sqrt((x*sin(c) + (sqrt(y^2+w^2)-5)*cos(c))^2 + z^2)-3)^2 + (x*cos(c) - (sqrt(y^2+w^2)-5)*sin(c))^2 = 1

4. Set up rotation from z->w, adjusting 't' will spin the tiger/ditorus around plane xy,

z = (z*sin(t)+w*cos(t))

w = (z*cos(t)-w*sin(t))

(sqrt((x*sin(c) + (sqrt(y^2+(z*cos(t)-w*sin(t))^2)-5)*cos(c))^2 + (z*sin(t)+w*cos(t))^2)-3)^2 + (x*cos(c) - (sqrt(y^2+(z*cos(t)-w*sin(t))^2)-5)*sin(c))^2 = 1

5. Convert equation to spherical coordinates, with adjustable rotational symmetry. Adjusting 'a' and/or 'b' will duplicate the number of inward-facing tori horizontally or vertically, adjusting 'd' will slide up/down in 4D,

x = (rho*cos(a*theta)*sin(b*phi))

y = (rho*sin(a*theta)*sin(b*phi))

z = (rho*cos(b*phi))

w = d

(sqrt(((rho*cos(a*theta)*sin(b*phi))*sin(c) + (sqrt((rho*sin(a*theta)*sin(b*phi))^2+((rho*cos(b*phi))*cos(t)-d*sin(t))^2)-5)*cos(c))^2 + ((rho*cos(b*phi))*sin(t)+d*cos(t))^2)-3)^2 + ((rho*cos(a*theta)*sin(b*phi))*cos(c) - (sqrt((rho*sin(a*theta)*sin(b*phi))^2+((rho*cos(b*phi))*cos(t)-d*sin(t))^2)-5)*sin(c))^2 = 1

Ranges:

-10 < rho < 10

0 < theta < pi

0 < phi < pi

XYZbox = -10 , 10

Adjustable Parameters:

a - changes rotation period of xy (1 < a < 4)

b - changes rotation period of xz (1 < b < 4)

c - changes general plane of rotation (0 < c < 1.57)

d - slides along 4D (-10 < d < 10)

t - rotates in 4D (0 < t < 1.57)


userPhilip

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Posted: 5/13/2015; 3:49:23 PM blueArrow

Here's a wild animation I made of the above function. Setting

a = 2

b = 2

c = animate from 0 to 6.28

d = 0

t = 0

http://i.imgur.com/1PhwIOv.gif


userPhilip

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Posted: 6/12/2015; 12:31:24 PM blueArrow

I made a cool gallery for a 4D hypercone, aka the spherical pyramid : http://imgur.com/a/vv7fG

And 4D conic sections, from a continuous surface: http://imgur.com/a/pgoAm


userPhilip

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Posted: 6/15/2015; 8:27:50 PM blueArrow

Here's a new tesseract gallery, using an amazing equation that produces tesseract slices in the form of 3D truss tube models:

http://hi.gher.space/forum/viewtopic.php?p=23876#p23876

The actual equation I used is:

(sqrt(abs(abs((x*sin(b)+a*cos(b))-(y*sin(c)+(x*cos(b)-a*sin(b))*cos(c)))+abs((x*sin(b)+a*cos(b))+(y*sin(c)+(x*cos(b)-a*sin(b))*cos(c)))-2(z*sin(d)+(y*cos(c)-(x*cos(b)-a*sin(b))*sin(c))*cos(d)))+abs(abs((x*sin(b)+a*cos(b))-(y*sin(c)+(x*cos(b)-a*sin(b))*cos(c)))+abs((x*sin(b)+a*cos(b))+(y*sin(c)+(x*cos(b)-a*sin(b))*cos(c)))+2(z*sin(d)+(y*cos(c)-(x*cos(b)-a*sin(b))*sin(c))*cos(d))))-3)^2+(sqrt(abs(abs((x*sin(b)+a*cos(b))-(y*sin(c)+(x*cos(b)-a*sin(b))*cos(c)))+abs((x*sin(b)+a*cos(b))+(y*sin(c)+(x*cos(b)-a*sin(b))*cos(c)))-2(z*cos(d)-(y*cos(c)-(x*cos(b)-a*sin(b))*sin(c))*sin(d)))+abs(abs((x*sin(b)+a*cos(b))-(y*sin(c)+(x*cos(b)-a*sin(b))*cos(c)))+abs((x*sin(b)+a*cos(b))+(y*sin(c)+(x*cos(b)-a*sin(b))*cos(c)))+2(z*cos(d)-(y*cos(c)-(x*cos(b)-a*sin(b))*sin(c))*sin(d))))-3)^2+(sqrt(abs(abs((x*sin(b)+a*cos(b))-(z*sin(d)+(y*cos(c)-(x*cos(b)-a*sin(b))*sin(c))*cos(d)))+abs((x*sin(b)+a*cos(b))+(z*sin(d)+(y*cos(c)-(x*cos(b)-a*sin(b))*sin(c))*cos(d)))-2(z*cos(d)-(y*cos(c)-(x*cos(b)-a*sin(b))*sin(c))*sin(d)))+abs(abs((x*sin(b)+a*cos(b))-(z*sin(d)+(y*cos(c)-(x*cos(b)-a*sin(b))*sin(c))*cos(d)))+abs((x*sin(b)+a*cos(b))+(z*sin(d)+(y*cos(c)-(x*cos(b)-a*sin(b))*sin(c))*cos(d)))+2(z*cos(d)-(y*cos(c)-(x*cos(b)-a*sin(b))*sin(c))*sin(d))))-3)^2+(sqrt(abs(abs((y*sin(c)+(x*cos(b)-a*sin(b))*cos(c))-(z*sin(d)+(y*cos(c)-(x*cos(b)-a*sin(b))*sin(c))*cos(d)))+abs((y*sin(c)+(x*cos(b)-a*sin(b))*cos(c))+(z*sin(d)+(y*cos(c)-(x*cos(b)-a*sin(b))*sin(c))*cos(d)))-2(z*cos(d)-(y*cos(c)-(x*cos(b)-a*sin(b))*sin(c))*sin(d)))+abs(abs((y*sin(c)+(x*cos(b)-a*sin(b))*cos(c))-(z*sin(d)+(y*cos(c)-(x*cos(b)-a*sin(b))*sin(c))*cos(d)))+abs((y*sin(c)+(x*cos(b)-a*sin(b))*cos(c))+(z*sin(d)+(y*cos(c)-(x*cos(b)-a*sin(b))*sin(c))*cos(d)))+2(z*cos(d)-(y*cos(c)-(x*cos(b)-a*sin(b))*sin(c))*sin(d))))-3)^2 = 0.025

XYZbox = -5 , 5

Corner First: b=1.0477 , c=0.9554 , d=0.785 , animate -5.5 < a < 5.5

Line First: b=1.5707 , c=0.9554 , d=0.785 , animate -5.5 < a < 5.5

Square First: b=1.5707 , c=1.5707 , d=0.785 , animate -5.5 < a < 5.5

Cube First: b=1.5707 , c=1.5707 , d=1.5707 , animate -5.5 < a < 5.5

Which goes to show you just how well CalcPlot can handle functions.


userPhilip

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Posted: 6/18/2015; 1:36:05 AM blueArrow

Here's a new gallery I just put up. It illustrates cube and hypercube shapes a bit differently, by using arrays of spheres and hyperspheres.

http://imgur.com/a/vAA3r


userPhilip

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Posted: 7/13/2015; 6:13:56 PM blueArrow

New animation gallery of multirotations of the 7D ((((II)I)I)((II)I)) :

http://imgur.com/a/62TFe


userPhilip

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Posted: 9/8/2015; 1:02:40 AM blueArrow

Here's the mathematical definitions of the construction operators, which can be remembered by the acronym STEMP : Spin, Taper, Extrude, Manifold, Product : O, >, I, (m), [m] . This is a breakdown on my procedural equation writing method, as based off my notation.

--


Spin / Bisecting Rotate : O

An n-dimensional shape rotates around an n-1 plane, into the n+1 dimension. This n-1 plane is a bisecting plane, that divides the shape in half. The n-1 plane acts like a stationary axle for the spinning motion of the rotation. Since the stationary plane has one dimension less than the whole shape, there will always be one axis left over: the non-stationary axis. This is the only variable we change in the starting shape's equation. All we have to do is choose a stationary plane, to isolate out the one and only non-stationary axis.

• Non-Stationary Axis : x_n

• Bisecting rotate into x_(n+1) :

x_n = √(x_n² + x_(n+1)²)

• If x_n is already in an n-sphere parameter, it becomes an n+1 sphere, without the extra sqrt term. Bisecting rotation of an n-sphere always leads to an (n+1)-sphere. We simply add more squared variables into the existing sqrt term.

√(x_n² + x_(n+1)²) = √(x_n² + x_(n+1)² + x_(n+2)²)

Some examples:

• Rotate Line I around origin, along X into Y, making circle IO

Line I : |x| = a

circle : √(x²+y²) = a

• Rotate Circle around X, along Y into Z, making Sphere IOO

Circle IO : √(x²+y²) = a

y² = y² + z²

Sphere : √(x²+y²+z²) = a

• Rotate Sphere IOO around XY, along Z into W, making Glome IOOO

Sphere IOO : √(x²+y²+z²) = a

z² = z² + w²

Glome : √(x²+y²+z²+w²) = a

• Rotate Triangle I> around Y, along X into Z, making Cone I>O

Triangle I> : ||x| + 2y| + |x| = a

x = √(x²+z²)

Cone : |√(x²+z²) + 2y| + √(x²+z²) = a , you can remove the absolute value around x

• Rotate Cone I>O around XZ, along Y into W, making Sphone I>OO = IOO>

Cone I>O : |√(x²+y²) + 2z| + √(x²+y²) = a

y² = y² + w²

Sphone : |√(x²+y²+w²) + 2z| + √(x²+y²+w²) = a

• Rotate Square II around Y, along X into Z, making Cylinder IIO

Square II : |x-y| + |x+y| = a

x = √(x²+z²)

Cylinder : |√(x²+z²)-y| + |√(x²+z²)+y| = a

• Rotate Cylinder IOI around XY, along Z into W, making Duocylinder IOIO

Cylinder IOI : |√(x²+y²) - z| + |√(x²+y²) + z| = a

z = √(z²+w²)

Duocylinder :|√(x²+z²) - √(z²+w²)| + |√(x²+z²) + √(z²+w²)| = a

• Rotate Cylinder IOI around XZ, along Y into W, making Spherinder IOOI

Cylinder IOI : |√(x²+y²) - z| + |√(x²+y²) + z| = a

y² = y²+w²

Spherinder : |√(x²+y²+w²) - z| + |√(x²+y²+w²) + z| = a

• Rotate Cube III around XY, along Z into W, making Cubinder IIIO

Cube IOI : ||x-y|+|x+y| - 2z| + ||x-y|+|x+y| + 2z| = a

z = √(z²+w²)

Cubinder : ||x-y|+|x+y| - 2√(z²+w²)| + ||x-y|+|x+y| + 2√(z²+w²)| = a

• Rotate Triangle Prism I>I around XY, along Z into W, making Cyltrianglinder I>IO

Triangle Prism I>I : |||x|+2y|+|x| - 2z| + |||x|+2y|+|x| + 2z| = a

z = √(z²+w²)

Cyltrianglinder : |||x|+2y|+|x| - 2√(z²+w²)| + |||x|+2y|+|x| + 2√(z²+w²)| = a

-----------


Taper : >

The taper operator m> will extend the base-shape m along the n+1 dimension, while scaling m to a point. This is the general pyramid construction operator. It works for all non-toroidal shapes without a hole. I can't seem to define a 4D toric pyramid, the toricone IO(O)>, quite yet. I always end up with a cone torus IO>(O), instead. The taper operator acts on the minor diameter only, rather than both.

• Equation for shape m : M

• Tapering m into x_(n+1) : |M + c*x_(n+1)| + M

c = the scaling constant, for the height of the pyramid. Since the variables in M are repeated twice, c should be set to equal out and match their numbers. Otherwise, you'll end up with tall/skinny cones and pyramids.

Some Examples:

• Taper Line I along Y, making Triangle I>:

Line I : |x| = M

m> = |M + 2y| + M

Triangle : ||x| + 2y| + |x| = a

• Taper Circle IO along Z, making Cone IO> :

Circle IO : √(x²+y²) = M

m> = |M + 2z| + M

Cone : |√(x²+y²) + 2z| + √(x²+y²) = a

• Taper Triangle I> along Z, making Tetrahedron I>>

Triangle I> : ||x|+2y|+|x| = M

m> = |M + 2z| + M

Tetrahedron : |||x|+2y|+|x| + 2z| + ||x|+2y|+|x| = a , the scaling for Z can be 2 instead of 4

• Taper Square II along Z, making Square Pyramid II>

Square II : |x-y|+|x+y| = M

m> = |M + 3z| + M

Pyramid : ||x-y|+|x+y| + 3z| + |x-y|+|x+y| = a , scaling for Z can be 3, looks better

• Taper Sphere IOO along W, making Sphone IOO> :

Sphere IOO : √(x²+y²+z²) = M

m> = |M + 2w| + M

Sphone : |√(x²+y²+z²) + 2w| + √(x²+y²+z²) = a

• Taper Cylinder IOI along W, making Cylindrone IOI> :

Cylinder IOI : |√(x²+y²)-z|+|√(x²+y²)+z| = M

m> = |M + 3w| + M

Cylindrone : ||√(x²+y²)-z|+|√(x²+y²)+z| + 3w| + |√(x²+y²)-z|+|√(x²+y²)+z| = a

• Taper Cone IO> along W, making Dicone IO>> :

Cone IO> : |√(x²+y²)+2z|+√(x²+y²) = M

m> = |M + 3w| + M

Dicone : ||√(x²+y²)+2z|+√(x²+y²) + 2w| + |√(x²+y²)+2z|+√(x²+y²) = a

-------------


Extrude : I

The extrude/extend operator mI will drag the starting shape m into the n+1 dimension. This is also the cartesian product of shape m times a line segment along x_(n+1). Makes a prism out the starting shape m.

• Equation for shape m : M

• Extend m along x_(n+1) : |M - c*x_(n+1)| + |M + c*x_(n+1)|

c = the scaling constant, for the height of the prism. Same rules as the taper apply.

Some Examples:

• Extend Line I along Y, making Square II:

Line I : |x| = M

MI = |M - y| + |M + y|

|x - y| + |x + y| = a , you can remove the abs around x

• Extend Circle IO along Z, making Cylinder IOI:

Circle IO : √(x²+y²) = M

mI = |M - z| + |M + z|

Cylinder : |√(x²+y²) - z| + |√(x²+y²) + z| = a

• Extend Triangle I> along Z, making Triangular Prism I>I

Triangle I> : ||x|+2y|+|x| = M

mI = |M - 2z| + |M + 2z|

Triangular Prism : |||x|+2y|+|x| - 2z| + |||x|+2y|+|x| + 2z| = a

• Extend Square II along Z, making Cube III

Square II : |x-y|+|x+y| = M

mI = |M - 2z| + |M + 2z|

Cube : ||x-y|+|x+y| - 2z| + ||x-y|+|x+y| + 2z| = a

• Extend Cone IO> along W, making Coninder IO>I:

Cone IO> : |√(x²+y²)+2z|+√(x²+y²) = M

mI = |M - 2w| + |M + 2w|

Coninder : ||√(x²+y²)+2z|+√(x²+y²) - 2w| + ||√(x²+y²)+2z|+√(x²+y²) + 2w| = a

• Extend Square Pyramid II> along W, making Square Pyraminder II>I

Square Pyramid II> : ||x-y|+|x+y|+3z|+|x-y|+|x+y| = M

mI = |M - 2w| + |M + 2w|

Pyraminder : |||x-y|+|x+y|+3z|+|x-y|+|x+y| - 4w| + |||x-y|+|x+y|+3z|+|x-y|+|x+y| + 4w| = a

• Extend Sphere IOO along W, making Spherinder IOOI:

Sphere IOO : √(x²+y²+z²) = M

mI = |M - w| + |M + w|

Spherinder : |√(x²+y²+z²) - w| + |√(x²+y²+z²) + w| = a

---------


Manifold : n(m) / Non-Bisecting Rotate (O)

For a simple non-intersecting rotation, the same rules apply as the bisecting rotation. In terms of choosing a stationary plane, and a non-stationary axis, we use a very similar variable change, to denote an embedding into the 1D edge of a disk. A basic non-bisecting sweeping m(O) starts with shifting the starting shape m along the non-stationary axis, by the value of b. Then, we replace the variable with an equation for a circle.

• Non-Stationary Axis : x_n

• Non-Bisecting Sweep along x_n into x_(n+1) :

x_n = (√(x_n² + x_(n+1)²) - b)

Some examples:

• Shift Circle IO by b along X, sweep around Y, along X into Z, making Torus IO(O)

Circle IO : √(x² + y²) = a

x = (√(x²+z²)-b)

Torus : √((√(x²+z²)-b)² + y²) = a

• Shift Triangle I> by b along X, sweep around Y, along X into Z, making Triangle Torus I>(O)

Triangle I> : ||x| + 2y| + |x| = a

x = (√(x²+z²)-b)

Triangle Torus : ||(√(x²+z²)-b)| + 2y| + |(√(x²+z²)-b)| = a , you have to keep the abs terms

• Shift Triangle I> by b along X, sweep around Y, along X into Z, making Square Torus II(O)

Square II : |x-y| + |x+y| = a

x = (√(x²+z²)-b)

Square Torus : |(√(x²+z²)-b)-y| + |(√(x²+z²)-b)+y| = a

-----------


Cartesian Product : n[m]

The cartesian product is when you multiply two solid shapes together, into one object. The simplest example is the square, as a product of two line segments. From the general product n[m] , the two shapes n and m are completely orthogonal, meaning they do not share the same variables. They are both perpendicular to each other. For the product n[m], we use all symbols for n and m. IO>[II>] is the 6D product of a cone and a square pyramid. IO>I[I>] is the 6D product of a coninder and a triangle, also equal to IO>[I>I]. IOO[I>>] is the 6D product of a sphere and tetrahedron.

The bisecting rotation of an n-prism nI, around the plane of n, is also the product of n and a circle. In other words, nIO == n[IO] . Extrude then rotate of n is equal to n x circle.

The double extrude of shape n, nII, is also the product with n x square: nII == n[II] . Rather than iterate two extrude operators, it's more efficient to express as product with a square, in the actual equation.

Note, however, that the product with a triangle in not equal to extrude then taper. In other words, m[I>] =/= mI> .

• Equation for n : N

• Equation for m : M

• Product of n[m] : |N - M| + |N + M|

Some examples:

• Product of Circle-XY times Circle-ZW, makes Duocylinder IOIO == IO[IO]. Also the bisecting rotation of cylinder IOI:

circle-xy = N = √(x²+y²)

circle-zw = M = √(z²+w²)

n[m] : |N - M| + |N + M|

Duocylinder : |√(x²+y²) - √(z²+w²)| + |√(x²+y²) + √(z²+w²)| = a

• Product of Triangle-XY times Circle-ZW, makes Cyltrianglinder I>IO == I>[IO]. Also the bisecting rotation of triangle prism I>I:

triangle-xy = N = ||x|+2y|+|x|

circle-zw = M = √(z²+w²)

n[m] : |N - 2M| + |N + 2M|

Cyltrianglinder : |||x|+2y|+|x| - 2√(z²+w²)| + |||x|+2y|+|x| + 2√(z²+w²)| = a

Note the scaling constant, c=2 for the circle-zw parameter. For n[m] products, the precise value of c is more critical to express a unit-proportioned shape.

• Product of Square-XY times Circle-ZW, makes Cubinder IIIO == II[IO]. Also the bisecting rotation of cube III, and extrusion of cylinder IOI:

square-xy = N = |x-y|+|x+y|

circle-zw = M = √(z²+w²)

n[m] : |N - 2M| + |N + 2M|

Cubinder : ||x-y|+|x+y| - 2√(z²+w²)| + ||x-y|+|x+y| + 2√(z²+w²)| = a

• Product of Square-XY times Square-ZW, makes Tesseract IIII == II[II]. Also the product of cube-xyz times line-w:

square-xy = N = |x-y|+|x+y|

circle-zw = M = |z-w|+|z+w|

n[m] : |N - 2M| + |N + 2M|

Tesseract : ||x-y|+|x+y| - |z-w|-|z+w|| + ||x-y|+|x+y| + |z-w|+|z+w|| = a


userPhilip

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Posted: 9/18/2015; 12:22:24 AM blueArrow

I rendered a new 8D hypertorus with a more advanced explore function than I've ever used. It's a modified form of the rotate/translate function that I wrote to turn the tesseract so that it will pass through a 3-plane vertex-first:

x = (x*sin(b) + a*cos(b))

y = (y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))

z = (z*sin(d) + (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*cos(d))

[xyzT] = ((z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d))*sin(t))

[xyzt] = ((z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d))*cos(t))

I put this function into a hypertorus equation, and found some really amazing and eye-catching things happen. Especially when we get to 7D and over. Here's the new gallery, with some other pics:

http://imgur.com/a/30wuA

First Animation:

• ((((X[xyzT]))(Y))((Z)[xyzt]))

(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + ((z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d))*sin(t))^2) -10)^2) -5)^2 + (sqrt((y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))^2) -5)^2) -2.5)^2 + (sqrt((sqrt((z*sin(d) + (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*cos(d))^2) -5)^2 + ((z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d))*cos(t))^2) -2.5)^2 = 1

Has very awesome morphs at a=5 , c=pi/4 , d=0 , t=0 , animate 0 < b < 2pi . Shows a momentary (((I)I)((I))) 2x1x4 square in two flip-flopped positions, with a very complex topology change with high level of symmetry.


Second Animation:

• ((((X[xyzT]))(Y))((Z[xyzt])))

(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + ((z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d))*sin(t))^2) -10)^2) -5)^2 + (sqrt((y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))^2) -5)^2) -2.5)^2 + (sqrt((sqrt((z*sin(d) + (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*cos(d))^2 + ((z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d))*cos(t))^2) -5)^2) -2.5)^2 = 1

Has awesome morphs analogous to above ((((X[xyzT]))(Y))((Z)[xyzt])) at a=5 , c=pi/4 , d=pi/2 , t=0 , animate 0 < b < 2pi . Shows a momentary (((II))((I))) 1x1x4x[R1 pair] column in two flip-flopped positions, between complex topo changes.


userPhilip

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Posted: 10/21/2015; 2:45:19 PM blueArrow

I put together a rather large gallery of 17 animations. These show various multirotations (double and triple) being done, to specific types of higher dimensional tori.

Imgur Gallery : http://imgur.com/a/ywa6B

Reddit /r/math link, where I describe in a little more detail the specifics and equations of the objects :

https://www.reddit.com/r/math/comments/3pggzy/multirotations_of_higher_dimensional_tori/


userPhilip

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Posted: 1/7/2016; 8:22:18 PM blueArrow

Here's some new animations of the elementary 4D tori, as rotating 3D slices, centered at origin : http://imgur.com/a/ZSTVs

The /r/math reddit link, with some more elaboration and better description : https://www.reddit.com/r/math/comments/3zilv4/rotating_four_dimensional_donuts/

Made using CalcPlot3D, with the following functions :

XYZ viewbox : -10 , +10

ctrl+shft+t = 120 , transparency

Adjust 't' from 0 to 2π for a 360 degree continuous spin

3D Torus Stuck in a 2-Plane

• 2-Plane : x=0

• Torus Rotation in 3D :

(sqrt((x*cos(t)+z*sin(t))^2 + y^2) -4)^2 + (x*sin(t)-z*cos(t))^2 = 4

• 2D Slice of Solid Torus Rotation, expressed as torus prism with aspect ratio of 1/10,000th 4D extension :

abs(sqrt((sqrt((z*sin(t))^2+y^2) -4)^2+(z*cos(t))^2)-10^4x)+abs(sqrt((sqrt((z*sin(t))^2+y^2) -4)^2+(z*cos(t))^2)+10^4x) = 4

• 4D Spheritorus Rotation

(sqrt((x*cos(t))^2 + y^2) -4)^2 + z^2 + (x*sin(t))^2 = 4

• 4D Torisphere Rotation

(sqrt((x*cos(t))^2 + y^2 + z^2) -4)^2 + (x*sin(t))^2 = 2

• 4D Tiger Rotation

(sqrt((x*cos(t))^2 + y^2) -4)^2 + (sqrt(z^2 + (x*sin(t))^2) -4)^2 = 2

• 4D 3-Torus Rotations

A :

(sqrt((sqrt((x*cos(t))^2 + y^2) -4)^2 + (x*sin(t))^2) -2)^2 + z^2 = 1

B:

(sqrt((sqrt((x*cos(t))^2 + y^2) -4)^2 + z^2) -2)^2 + (x*sin(t))^2 = 1

C:

(sqrt((sqrt(x^2 + y^2) -4)^2 + (z*cos(t))^2) -2)^2 + (z*sin(t))^2 = 1


userPhilip

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Posted: 3/3/2016; 3:15:06 PM blueArrow

Cool new gallery of gifs, made using calcplot, that illustrate where a 4th dimension would point to, from a 3D perspective:

http://imgur.com/a/XZpBP

The functions I used for each gif:

In all of them, for the rotate function, use 'a' to slide up/down along the 3rd/4th axis ; use 'b' to change the angle, from 0 to pi/2 for a 90 degree turn. The 8 objects in a row are defined as the solutions of even higher dimensional donuts. Start by embedding your original object into the surface of a much larger 3-torus. Set all but one particular variable to 0 in the 3-torus component, and you'll get 8 copies of the smaller embedded object stacked along a line. What I did was use the long line of 8 as a single, static thing, to scan in 2D/3D. The spheres and tori are intersections of 2 different 6D tori. The 8 copies of a 4D object are intersections of different 7D tori.

-


Line of 8 Spheres

• 3D View of 8 Spheres :

(sqrt((sqrt((sqrt((y*cos(b)+(x+a)*sin(b))^2) -8)^2) -4)^2) -2)^2 + (z+c)^2 + (y*sin(b)-(x+a)*cos(b))^2 = 2

• 2D Slice of 8 spheres, in 3D environment (the darker turquoise circles) :

abs(sqrt((sqrt((sqrt((sqrt((y*cos(b)+a*sin(b))^2) -8)^2) -4)^2) -2)^2 + (z-c)^2 + (y*sin(b)-a*cos(b))^2) - dx) + abs(sqrt((sqrt((sqrt((sqrt((y*cos(b)+a*sin(b))^2) -8)^2) -4)^2) -2)^2 + (z-c)^2 + (y*sin(b)-a*cos(b))^2) + dx) = 2.75

c = 7 : separation distance between the 3D and 2D view ; you can omit this parameter, if you only want the 8 spheres, just don't plot the 2D slice function

d = 20 : scaling coefficient to collapse size of the 4D extension, i.e. height of the cylinders into circles

-


• Line of 8 Hyperspheres

(sqrt((sqrt((sqrt((x*cos(b)+a*sin(b))^2) -8)^2) -4)^2) -2)^2 + (x*sin(b)-a*cos(b))^2 + y^2 + z^2 = 2

-


Column of 8 Toruses

• 3D Perspective :

(sqrt((sqrt((sqrt((y*cos(b)+(x+a)*sin(b))^2) -8)^2) -4)^2) -2)^2 + (sqrt((z+c)^2 + (y*sin(b)-(x+a)*cos(b))^2) -2)^2 = 1

• 2D Slices in 3D environment, the darker turquoise image :

abs(sqrt((sqrt((sqrt((sqrt((y*cos(b)+a*sin(b))^2) -8)^2) -4)^2) -2)^2 + (sqrt((z-c)^2 + (y*sin(b)-a*cos(b))^2) -2)^2) - dx) + abs(sqrt((sqrt((sqrt((sqrt((y*cos(b)+a*sin(b))^2) -8)^2) -4)^2) -2)^2 + (sqrt((z-c)^2 + (y*sin(b)-a*cos(b))^2) -2)^2) + dx) = 1.75

c = 6 : separation distance between the 3D and 2D view

d = 20 : scaling coefficient to collapse size of 4D extension

-


•Column of 8 T3 's Rotation type 1

(sqrt((sqrt((sqrt((x*cos(b)+a*sin(b))^2) -8)^2) -4)^2) -2)^2 + (sqrt((sqrt((x*sin(b)-a*cos(b))^2 + y^2) -4)^2 + z^2) -2)^2 = 1

-


• Column of 8 T3 's rotation type 2

(sqrt((sqrt((sqrt((x*cos(b)+a*sin(b))^2) -8)^2) -4)^2) -2)^2 + (sqrt((sqrt(y^2 + z^2) -4)^2 + (x*sin(b)-a*cos(b))^2) -2)^2 = 1

-


• Row of 8 Tigers

(sqrt((sqrt((sqrt((sqrt((x*cos(b)+a*sin(b))^2) -16)^2) -8)^2) -4)^2 + y^2) -2)^2 + (sqrt((x*sin(b)-a*cos(b))^2 + z^2) -2)^2 = 1

-


• Column of 8 Torispheres

(sqrt((sqrt((sqrt((x*cos(b)+a*sin(b))^2) -8)^2) -4)^2) -2)^2 + (sqrt((x*sin(b)-a*cos(b))^2 + y^2 + z^2) -3)^2 = 1

-


• Column of 8 Spheritoruses

(sqrt((sqrt((sqrt((x*cos(b)+a*sin(b))^2) -8)^2) -4)^2) -2)^2 + (sqrt((x*sin(b)-a*cos(b))^2 + y^2) -2)^2 + z^2 = 1


userPhilip

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Posted: 3/21/2016; 3:34:19 PM blueArrow

The slices of a circle, torus, 3-torus, 4-torus, and 5-torus

3 separate images:

http://imgur.com/gallery/5boei

single, full poster image

http://imgur.com/gallery/NXnBqt5


userPhilip

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Posted: 10/2/2016; 1:04:14 PM blueArrow

I made my first projection animation of a Klein Bottle. Instead of accumulating screenshots, and making a gif, I just ran calcplot in realtime and used a screencapture recorder.

https://www.youtube.com/watch?v=aobRi4KQcKA&feature=youtu.be

This animation was made with the parametric equation:

x = (3 + sin(v))cos(u)

y = (3 + sin(v))sin(u)

z = cos(v)cos(u/2)

w = cos(v)sin(u/2)

rotate on plane xw:

x = (x)*cos(t) + (w)*sin(t)

y = y

z = z

w = (x)*sin(t) - (w)*cos(t)

becomes,

x = ((3 + sin(v))cos(u))*cos(t) + (cos(v)sin(u/2))*sin(t)

y = (3 + sin(v))sin(u)

z = cos(v)cos(u/2)

w = ((3 + sin(v))cos(u))*sin(t) - (cos(v)sin(u/2))*cos(t)

Project on plane xyz:

x = ((3 + sin(v))cos(u))*cos(t) + (cos(v)sin(u/2))*sin(t)

y = (3 + sin(v))sin(u)

z = cos(v)cos(u/2)

w = 0

Vary parameter t between 0 and 2π to rotate on plane xw

-----------------


For exploring the cross sections of a mobius tube (klein bottle) , use the implicitized polynomial:

(x^2 +y^2 +z^2 +w^2 -a^2 -b^2)^2 -4a^2(b^2 -z^2 -w^2) = 0

which can also be represented as

(sqrt(x^2 +y^2) -a)^2 +z^2 +w^2 = b^2

rotating on plane xw, set w=d :

x = (x*cos(t) + d*sin(t))

w = (x*sin(t) - d*cos(t))

becomes a movable 3D cross section:

((x*cos(t)+d*sin(t))^2 +y^2 +z^2 +(x*sin(t)-d*cos(t))^2 -a^2 -b^2)^2 -4a^2(b^2 -z^2 -(x*sin(t)-d*cos(t))^2) = 0

set a = 3 ; b = 1

Exploring this function using ‘d’ to translate along w , ’t’ to rotate on xw , you will see all of the slices of S2 x S1 , the spheritorus, symbolized by ((II)II) in toratope notation:

Rotating at origin : http://i.imgur.com/9AZPAua.mp4

Passing through 3D at various angles : http://i.imgur.com/cYzvzEl.mp4


userPhilip

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Posted: 10/6/2016; 5:11:06 PM blueArrow

Here's a 3-torus graphed and explored in real-time (a major advantage that calcplot has over other rendering programs):

https://www.youtube.com/watch?v=VnYJWkV9CCs

I used the equation:

(sqrt((sqrt(x^2+(y*sin(b)+a*cos(b))^2)-4)^2+(z*sin(c)+(y*cos(b)-a*sin(b))*cos(c))^2)-2)^2+(z*cos(c)-(y*cos(b)-a*sin(b))*sin(c))^2 = 1

Using the adjustable sliders:

a : translate along 4-space

b : rotate on plane yz

c : rotate on plane zw

Combining angles b and c allows you to see double-tilted scans of an object, which are even more cool than single-tilt.


userPhilip

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Posted: 10/9/2016; 10:12:42 PM blueArrow

3D Slices of a Tiger in Real Time :

https://www.youtube.com/watch?v=lnQ4Zfj18Og

The equation used in the video can be derived through the following process:

• Start with a circle of radius 1, on plane xz:

x^2 + z^2 = 1

• Translate along x-axis by 2 units:

(x - 2)^2 + z^2 = 1

• Revolve around the z-axis on plane xy into 3D, to create a torus:

(sqrt(x^2 + y^2) -2)^2 + z^2 = 1

• Translate the torus along z-axis by 2 units:

(sqrt(x^2 + y^2) -2)^2 + (z - 2)^2 = 1

• Revolve around the xy-plane, on plane zw into 4D, to create a tiger ((II)(II)) :

(sqrt(x^2 + y^2) -2)^2 + (sqrt(z^2 + w^2) -2)^2 = 1

More generally,

(sqrt(x^2 + y^2) -R1)^2 + (sqrt(z^2 + w^2) -R2)^2 = r^2

R1 ≥,≤ R2 > r

• To set up a rotate/translate equation, replace:

x = (x*cos(b) - a*sin(b))

w = (x*sin(b) + a*cos(b))

which becomes (actual value of r is 0.75^2 ):

(sqrt((x*cos(b) - a*sin(b))^2 + y^2) -2)^2 + (sqrt(z^2 + (x*sin(b) + a*cos(b))^2) -2)^2 = 1

set the ranges:

• Translate ana/kata in 4-space:

-4 < a < 4

• Rotate 90° on plane xw:

0 < b < π/2 (or 2π for 360°)

• When a=0 and b=π/4 (45°) , we get a structure that I call the Tiger Cage , which is a good image to remember, when thinking of the tiger. Some slices of 5D, 6D, 7D, etc, tori can result in arrays of multiple cages.

--------


So, we see that just like a 3-torus, a tiger is also a surface revolution of a torus into 4D. Instead of shifting a torus laterally along x or y, we shift vertically, along z. That's why we get the vertically separated pair of tori, when we slice into 3D. The two disjoint tori are the intercepts of the 4D ring.

Another way of understanding the tiger is to think of embeddings, where you replace every point of a surface with a whole other, smaller surface. The tiger can be made by embedding a circle of radius 1 into a flat 2-torus of radii 2 and 2. The flat 2-torus (clifford torus) is a fundamental polygon, and the tiger is a good way to visualize it (if you set the minor diameter r to very small sizes).

Another curious feature of the tiger, are the two major diameters R1 and R2. If we define a 'major diameter' as one that can be freely adjusted to any large value, and maintain no self-intersection, then a tiger has two of them. A 3-torus has just one major, a secondary, and a minor.


This page was last updated: Dec. 21, 2016

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