## Renderings of Open Toratopes

Discussion of shapes with curves and holes in various dimensions.

### Renderings of Open Toratopes

Apologies for posting a 6D shape first, but I recently checked this one out. It's the bi-toroidal prism, also equal to a duocylindric duotorus.

Exploration of the 6D Duocylindrical Duotorus / Bi-Toroidal Prism ((II)I)((II)I)
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((II)I)((II)I) - open toratope notation
IO(O)[IO(O)] - cartesian product of 2 orthogonal tori, (torus*torus)-prism, B^2 x S^1 x B^2 x S^1
IOIO[(O)(O)] - duocylinder bundle over the clifford torus, B^2 x B^2 x C^2

A few ways to build [((II)I)((II)I)] :

• I - 1D Line

|x| = a

• IO - 2D Circle, bisecting Line rotate around origin, along x into y

√(x²+y²) = a

• IO(O) - 3D Torus, shift Circle by -a along x, sweep along x into z

√((√(x²+z²)-a)² + y²) = b

√((√(x²+y²)-a)² + z²) = b

• IO(O)I - 4D Torinder, extend Torus along w

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

• IO(O)IO - 5D Cyltorinder, bisecting rotate Torinder around xyz, along w into v

|√((√(x²+y²)-a)²+z²) - √(w²+v²)| + |√((√(x²+y²)-a)²+z²) + √(w²+v²)| = b

• IO(O)[IO(O)] - 6D Bi-Toroidal Prism, shift Cyltorinder by -b along w, sweep along w into u

|√((√(x²+y²)-a)²+z²) - √((√(w²+u²)-b)²+v²)| + |√((√(x²+y²)-a)²+z²) + √((√(w²+u²)-b)²+v²)| = c

|√((√(x²+y²)-a)²+z²) - √((√(w²+v²)-b)²+u²)| + |√((√(x²+y²)-a)²+z²) + √((√(w²+v²)-b)²+u²)| = c

• I - 1D Line

|x| = a

• IO - 2D Circle, bisecting rotate Line around origin, along x into y

√(x²+y²) = a

• IOI - 3D Cylinder, extend Circle along z

|√(x²+y²) - z| + |√(x²+y²) + z| = a

• IOIO - 4D Duocylinder, bisecting rotate Cylinder around xy, along z into w

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

• IOIO(O) - 5D Duocylindric Torus, shift Duocylinder by -a along x, sweep along x into v

|√((√(x²+v²)-a)²+y²) - √(z²+w²)| + |√((√(x²+v²)-a)²+y²) + √(z²+w²)| = b

|√((√(x²+y²)-a)²+z²) - √(w²+v²)| + |√((√(x²+y²)-a)²+z²) + √(w²+v²)| = b

• IOIO[(O)(O)] - 6D Duocylindric Duotorus, shift Duocylindric Torus by -b along w, sweep along w into u

|√((√(x²+y²)-a)²+z²) - √((√(w²+u²)-b)²+v²)| + |√((√(x²+y²)-a)²+z²) + √((√(w²+u²)-b)²+v²)| = c

|√((√(x²+y²)-a)²+z²) - √((√(w²+v²)-b)²+u²)| + |√((√(x²+y²)-a)²+z²) + √((√(w²+v²)-b)²+u²)| = c

|√((√(x²+y²)-R1a)²+z²) - √((√(w²+v²)-R1b)²+u²)| + |√((√(x²+y²)-R1a)²+z²) + √((√(w²+v²)-R1b)²+u²)| = Rminor

abs(sqrt((sqrt(x^2+y^2)-R1a)^2+z^2) - sqrt((sqrt(w^2+v^2)-R1b)^2-u^2)) + abs(sqrt((sqrt(x^2+y^2)-R1a)^2+z^2) + sqrt((sqrt(w^2+v^2)-R1b)^2+u^2)) = Rm

abs(sqrt((sqrt(x^2+y^2)-3)^2+z^2) - sqrt((sqrt(w^2+v^2)-3)^2-u^2)) + abs(sqrt((sqrt(x^2+y^2)-3)^2+z^2) + sqrt((sqrt(w^2+v^2)-3)^2+u^2)) = 1

3D Midsections

XYZbox = -7 / +7

• ((II))((I)) : Vertical stack of 2 square tori , ZVU=0
abs((sqrt(x^2+y^2) -3)^2 + 0^2 - (sqrt(z^2+0^2) -3)^2 - 0^2) + abs((sqrt(x^2+y^2) -3)^2 + 0^2 + (sqrt(z^2+0^2) -3)^2 + 0^2) = 3

• ((I)I)((I)) : 4 cylinders in 2x2 square array , YVU=0
abs((sqrt(x^2+0^2) -3)^2 + y^2 - (sqrt(z^2+0^2) -3)^2 - 0^2) + abs((sqrt(x^2+0^2) -3)^2 + y^2 + (sqrt(z^2+0^2) -3)^2 + 0^2) = 3

3D Explore Functions

• ((IY))((Iy)) : Single Rotate between stacks of square tori
abs((sqrt(x^2+(y*sin(a))^2) -3)^2 + 0^2 - (sqrt(z^2+(y*cos(a))^2) -3)^2 - 0^2) + abs((sqrt(x^2+(y*sin(a))^2) -3)^2 + 0^2 + (sqrt(z^2+(y*cos(a))^2) -3)^2 + 0^2) = 3

• ((XY)z)((Zx)y) : Triple Rotate between stacks of sq tori, four cylinders in square, and empty holes
abs((sqrt((x*sin(a))^2+(y*sin(b))^2) -3)^2 + (z*cos(c))^2 - (sqrt((z*sin(c))^2+(x*cos(a))^2) -3)^2 - (y*cos(b))^2) + abs((sqrt((x*sin(a))^2+(y*sin(b))^2) -3)^2 + (z*cos(c))^2 + (sqrt((z*sin(c))^2+(x*cos(a))^2) -3)^2 + (y*cos(b))^2) = 3
—— Cage of Bi-toroidal prism at a=0.785 ; b,c=1.57 // Direct analogue to Cage of Duotorus Tiger (((IO))((IO)))
—— 0 < a,b,c < 1.57

• ((Xz)Y)((Zx)y) : Triple Rotate between duocylinder sections in square array and empty holes
abs((sqrt((x*sin(a))^2+(z*cos(c))^2) -3)^2 + (y*sin(b))^2 - (sqrt((z*sin(c))^2+(x*cos(a))^2) -3)^2 - (y*cos(b))^2) + abs((sqrt((x*sin(a))^2+(z*cos(c))^2) -3)^2 + (y*sin(b))^2 + (sqrt((z*sin(c))^2+(x*cos(a))^2) -3)^2 + (y*cos(b))^2) = 4
—— Very interesting structure at a,c=1.157 ; b=0.785
—— 0 < a,b,c < 1.57

• ((AC)c)((Ia)) : Dual Translate+Rotate
abs((sqrt((x*sin(b)+a*cos(b))^2+(y*sin(d)+c*cos(d))^2) -3)^2 + (y*cos(d)-c*sin(d))^2 - (sqrt(z^2+(x*cos(b)-a*sin(b))^2) -3)^2 - 0^2) + abs((sqrt((x*sin(b)+a*cos(b))^2+(y*sin(d)+c*cos(d))^2) -3)^2 + (y*cos(d)-c*sin(d))^2 + (sqrt(z^2+(x*cos(b)-a*sin(b))^2) -3)^2 + 0^2) = 3
—— -7 < a,c < +7
—— 0 < b,d < 1.57

And some neat pics:

Villarceau Section of Oblique Duocylinder Cut

This is where the four duocylinders in a square array have been turned, making the 4 crinds in a square. Then, rotate in two separate ways to make a tetra-tangent Villarceau section of those oblique structures.

Cage of Bi-Toroidal Prism

The 45 degree oblique structure, between the two vertical stacks of square tori. The square torus is actually a 3D section of a Cyltorinder / Duocylindric Torus, where this 6D shape intercepts as two of them in a column.

Which is directly related to the Cage of Duotorus Tiger. By spherating this structure of the bi-toroidal prism, we'll empty out the flat 2-surfaces, and expand the sharp edges with a circle in every point. The result is this:

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### Re: Renderings of Open Toratopes

The 4D Spherinder

I - 1D Line : extend Point along 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

IOOI - 4D Spherinder : extend Sphere along W

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

3D General Cross Section Equation:

abs(sqrt(x^2+y^2+(z*sin(b)+a*cos(b))^2)-(z*cos(b)-a*sin(b))) + abs(sqrt(x^2+y^2+(z*sin(b)+a*cos(b))^2)+(z*cos(b)-a*sin(b))) = 7

set range:

-6 < a < 6

0 < b < 1.57

Adjust 'a' to slide in 4D up/down. a=0 is the middle.

Adjust 'b' to rotate in 4D. b=0 is the cylinder , b=1.57 is the sphere slice.

Rotating a cylinder while stuck in a 2D plane

What's important here, is what's happening on the big green sheet. We see a square-slice morph into a circle-slice, by rotating a 3D object.

Rotaing 2D Cross Section of a Cylinder

Confined to just the 2D square, we'll see only lines, morphing from square to circle. This is also what happens when we use a 3D plane to image a 4D object. To explore a 3D cylinder in 2D, see here: https://www.desmos.com/calculator/otpeykbx8g

Rotating 3D Cross Section of a 4D Spherinder

Using a 3-plane to image a 4D object will make only 3D objects. Where again, there's more than meets the eye. If we turn this shape in 4D, the cylinder morphs into a sphere, identical to the cylinder rotation.

A spherinder is the spherical prism, made by extending a sphere into a fourth dimension of space. It's just like a 3D cylinder, which has two flat circle endcaps, joined by a hollow tube. A spherinder has two flat spherical endcaps, joined by a hollow hyper-tube.

A cylinder's hollow tube is a 2D sheet, connecting the 1D edge of two circles. The hyper-tube is a 3D surface, connecting the 2-surface of two spheres together. Imagine a cylinder's tube punching out a circular piece of 2D space. In the same exact way, a spherinder's hyper-tube can punch out a spherical piece of 3D space.

Passing a Cylinder through 2D

Showing what a cylinder looks like when passing through, at different angles of a 90 degree turn. The cylinder is getting scanned just like a CAT scanner, imaging a 3D object in 2D.

2D Scans of a 3D Cylinder

A 2D being confined to this square will see only the 1D line-slices of the cylinder. Passing through at 0 deg makes a line expand to a square, then collapse to line. The square is said to have "circular height". This is from slicing through the circle-part of the cylinder, into line segments, leaving the line extension alone.

At 90 deg, we see a circle appear, remain still and unchanged, then suddenly disappear. This is from slicing the line extension into points, leaving the circle-part alone. From this, we can see how a cylinder has infinite circles stuck into every point within a line segment.

3D Scans of a 4D Spherinder

Using the same technique as above, here are the 3D images of scanning a 4D sphere prism. At 0 deg, we see the circle part of the cylinder expand and contract. This is from slicing through the sphere part of the spherinder, leaving the line extension alone. Slicing a sphere into a circles looks like this .

At 90 deg, we see a sphere appear, and remain unchanged, then disappear. This is from slicing through the line extension part into points, leaving the sphere-part alone. What this does, is shows us how a sphere prism has infinite spheres stuck into every point along a line segment.

Another good thing to bring up is how a spherinder is related to two different 4D hyperdonuts. If we bent a cylinder into a circle, and glued the circle endcaps together, we'd get a torus (donut). Now, by bending a spherinder into a circle, and gluing its sphere endcaps, we'd get the spheritorus .

If we snipped out the circles of a cylinder, we'd get a hollow tube. Now, take this tube and bend back one of the open edges, and roll it back onto itself, and connect the ends. We'd get another torus. Using the same sock-rolling method, we can snip out the spheres of a spherinder. We'd get the hollow hyper-tube, as a line extension of the surface of two spheres. Now take this hypertube and roll it back, and we turn the line extension into a circular extension of a sphere's 2-surface : the Torisphere
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### Re: Renderings of Open Toratopes

Here's the Sphone:

In order to understand the slices of a hypercone, we need to see what slices of a cone look like. The important part here is what happens on the green sheet. By turning a 3D shape 90 degrees, we see a circle slice transform into a triangle slice. This happens by cutting through different ways.

This is what it looks like after removing the 3D part out. Here's an interactive explorer of a 3D cone, in 2D : https://www.desmos.com/calculator/w3xptfnyhb . Adjusting 'a' will slide up/down in 3D. Adjusting 'b' will turn 90 degrees, back and forth. While doing this, try to imagine the rest of the cone, and the angle it's being sliced.

This is a 3D view of turning the hypercone back and forth. We see a sphere transform into a cone. There's nothing magical going on here, just a more extensive object being sliced at different angles, like a 3D cone in 2D.

Here is a 3D view of passing a cone through 2D at five different angles. Again, the important part is what happens on the green sheet. If you follow closely, you can see a large circle suddenly appear, and shrink. At the 90 degree scan, we see a flat dome grow into a triangle, then smoothly collapse again.

Same as above, but removing the 3D part out of the way. Now it's easier to see what's going on. You could say the lines are moving as predetermined by the surface of the cone.

Here we are at last: the 3D multi-angle scans of a hypercone! You may notice the similarities between this and the 3D cone. They are closely related, after all. In fact, you can get slices of a hypercone, just by rotating any slice of a cone. This is because a hypercone can also be made by rotating a cone into 4D.
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### Re: Renderings of Open Toratopes

And some 4D conic sections, with 3D analogy:

Rotating the conic surface 360 degrees, while sliced by a 2D plane. This single surface can represent circles, ellipses, parabolas, and hyperbolas. Graphed by the equation : x² + y² = z²

What we will see in only 2D. This are the intersections on the green sheet, from the above animation.

This one comes out as an inversion of the 2-sheet. This surface can also be made from a surface revolution of a hyperbola. Graphed by the equation : x² + y² = z² + w²

In visual terms, this is the closer analogy to a 3D conic surface. It can also be made by a surface revolution of the first two animations. Graphed by the equation : x² + y² + z² = w²
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### Re: Renderings of Open Toratopes

I made a new tesseract gallery, using the circle over 2-frames of tesseract equation:

From this perspective, the green sheet is Flatland. Here, we see a 3D cube being dropped through it at different angles. The 3D cube is a higher dimensional object compared to the Flatlanders, that are confined to their 2D world.

This is a top-down perspective of what a 2D being will see. The 3D part of the cube has been removed. Notice how the corner scan expands in all two directions available.

3D view of a Hypercube passing through Spaceland. Now WE are the Flatlander beings, by comparison. A 4D tesseract is a higher dimensional object to us, which means we cannot see all of it, like a 4D being can. This is what it looks like being dropped through a flat 3D sheet. This is actually a hollowed out hypercube, not a solid one. The cube-slice starts off solid, then empties out, to show the boundary squares on the outside. A hypercube can be scanned cube, square, line and corner first. The cube-first is the additional one we don't normally think about or expect.

A hypercube can be made by extending a cube into 4D, like a prism. So, when the square faces of a cube get extended, they turn into cubes themselves. At the very and middle, our 3D space is slicing from one end to the other, making the rectangular version of that cube slice. At this depth, we see the boundary squares appear as filled-in squares. We also see the same filled in squares when it begins to pierce 3D.

Same as above, this is the extended into prism version of a cube scan. This means the original corners of the starting cube also get extended, into lines. When we pass a hypercube through one of these lines first, we see the extended version of that corner scan of a cube : a triangle prism. It then morphs into an extended version of the middle slice of a cube : a hexagon prism.

Remember how the corner scan of a cube make a triangle? A hypercube slice in the same exact way makes the 3D version of a triangle : a tetrahedron. Notice how it expands in all three directions available. Once the hypercube reaches a certain depth at its other corners, we see the tetrahedron begin to truncate, or flatten its corners, as it snipped off. The very middle is an octahedron slice, just like the hexagon from a cube. The very moment we see the dot appear and grow is when the corner pierces our 'flat' 3D world. So, you have to ask yourself: Where is this thing actually coming from, and where does it go? Well, a mysterious, extra fourth dimension, or course!
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### Re: Renderings of Open Toratopes

Another cool gallery of cube and hypercube shaped things. The 4x4x4x4 tesseract array of glomes is the 12D toratope (((II)I)((II)I)((II)I)((II)I)) , passing through as if it were just a tesseract.

A Cube of Spheres passing through 2D, square first

2D view only, of the 4x4x4 cube, passing square first

A Cube of Spheres passing through 2D, line first

2D view only, of the 4x4x4 cube, passing line first

A Cube of Spheres passing through 2D, corner first

2D view only, of the 4x4x4 cube, passing corner first

A Hypercube of Hyperspheres passing through 3D, Cube First

A Hypercube of Hyperspheres passing through 3D, Square First

A Hypercube of Hyperspheres passing through 3D, Line First

A Hypercube of Hyperspheres passing through 3D, Corner First

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### Re: Renderings of Open Toratopes

I think it would be better if the number of spheres per edge was odd -- this way, the halfway cut for corners is empty.

A hybrid approach might be to scan two wireframe tesseracts with different edge lengths, one inside the other...
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### Re: Renderings of Open Toratopes

I think it would be better if the number of spheres per edge was odd -- this way, the halfway cut for corners is empty.

I was thinking the same thing, actually. Odd numbers would make the truncated tetrahedral arrays more apparent. But, I'm not sure about how the octahedral midsection would look, or if it would even exist.

A hybrid approach might be to scan two wireframe tesseracts with different edge lengths, one inside the other...

Strangely enough, I was also thinking this, too! In a similar approach, it occurred to me that I could make a true projection of the tesseract, and maybe have a rotation programmed into it. This can be done with a product of 24 narrow cylinders, all joined as a piecewise function. All I would need is the maths for a tesseract's vertices.

Using a general equation for a single cylinder, with full translate/rotate in 3d could make this feasible. Maybe. I'd be even more interested in trying it with something more exotic, like a pyramid prism. It would be rotated to its pyramid, cube, and triangle prism faces, with full continuity.
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