## Implicit Definition of Other Shapes

Higher-dimensional geometry (previously "Polyshapes").

### Re: Implicit Definition of Other Shapes

Explored the 5D Contrianglinder IO>[I>] tonight. Some really awesome shape morphs in this one! All of this is completely new to me, I've never seen anything like it. In addition to toratopes, they sure do make really interesting morphs and intercepts. No gifs yet, maybe tomorrow.

Here's a CalcPlot3D exploration script. This is the rotate function at the bottom : 5D Contrianglinder.txt

[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 , other orientation
[0,0.785] - Square Pyramid II> , unanticipated oblique cut at 45 deg between both I>I mid-cuts

IO>[I>] : 5D Contrianglinder Explore Functions
------------------------------------------------------------
I = 2*abs(x) - a
I> = abs(abs(x) - 2y) + abs(x) - a
I>O = abs(sqrt(x^2 + y^2) - 2z) + sqrt(x^2 + y^2) - a^2
I>O[I>] = abs(abs(sqrt(x^2+y^2) - 2z) + sqrt(x^2+y^2) - abs(abs(w)-2v) - abs(w)) + abs(abs(sqrt(x^2+y^2) - 2z) + sqrt(x^2+y^2) + abs(abs(w)-2v) + abs(w)) - a

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

3D Cross Sections of IO>[I>] along plane XYZWV
-----------------------------------------------------------

• >[I>] : Triangle Prism I>I , cancel out plane XY
abs(abs(sqrt(a^2+b^2) - 2x) + sqrt(a^2+b^2) - abs(abs(y)-2z) - abs(y)) + abs(abs(sqrt(a^2+b^2) - 2x) + sqrt(a^2+b^2) + abs(abs(y)-2z) + abs(y)) - 10
--- a,b : height has I> || I>I with circular symmetry

• I[I>] : Triangle Prism I>I , cancel out plane YZ
abs(abs(sqrt(x^2+a^2) - 2b) + sqrt(x^2+a^2) - abs(abs(y)-2z) - abs(y)) + abs(abs(sqrt(x^2+a^2) - 2b) + sqrt(x^2+a^2) + abs(abs(y)-2z) + abs(y)) - 10
--- a : z-height has circ symm
--- b : z-height has half height triangle symm

• I>[>] : Triangle Prism I>I , cancel out plane YW
abs(abs(sqrt(x^2+a^2) - 2y) + sqrt(x^2+a^2) - abs(abs(b)-2z) - abs(b)) + abs(abs(sqrt(x^2+a^2) - 2y) + sqrt(x^2+a^2) + abs(abs(b)-2z) + abs(b)) - 10
--- a : y-width has smoothing collapse to line
--- b : z-height has full height triangle symm

• I>[I] : Triangle Prism I>I , cancel out plane YV
abs(abs(sqrt(x^2+a^2) - 2y) + sqrt(x^2+a^2) - abs(abs(z)-2b) - abs(z)) + abs(abs(sqrt(x^2+a^2) - 2y) + sqrt(x^2+a^2) + abs(abs(z)-2b) + abs(z)) - 10
--- a : y-width has smoothing collapse to line
--- b : z-height has half height triangle symm

• IO[>] : Cylinder IOI , cancel out plane ZW
abs(abs(sqrt(x^2+y^2) - 2a) + sqrt(x^2+y^2) - abs(abs(b)-2z) - abs(b)) + abs(abs(sqrt(x^2+y^2) - 2a) + sqrt(x^2+y^2) + abs(abs(b)-2z) + abs(b)) - 10
--- a : I || IOI , line atop cylinder symmetry
--- b : z-height has half-height triangular symmetry

• IO[I] : Cylinder IOI , cancel out plane ZV
abs(abs(sqrt(x^2+y^2) - 2a) + sqrt(x^2+y^2) - abs(abs(z)-2b) - abs(z)) + abs(abs(sqrt(x^2+y^2) - 2a) + sqrt(x^2+y^2) + abs(abs(z)-2b) + abs(z)) - 10
--- a : I || IOI , line atop cylinder with half-height triangle symm
--- b : IO || IOI , circle atop cylinder with half-height triangle symm

• IO>[] : Cone IO> , cancel out plane WV
abs(abs(sqrt(x^2+y^2) - 2z) + sqrt(x^2+y^2) - abs(abs(a)-2b) - abs(a)) + abs(abs(sqrt(x^2+y^2) - 2z) + sqrt(x^2+y^2) + abs(abs(a)-2b) + abs(a)) - 10
--- a,b : cone has line symmetry

• I>I , IOI , IO> Rotation Function
----------------------------------------------
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>
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### Re: Implicit Definition of Other Shapes

And, here's the gif of contrianglinder. I recently found more rotation morphs, when parameter 'b' is set to 0 < b < 3.14 Using the rotate function, these are the updated a,b positions:

[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

Some neat things to do are set

0.785 < b < 2.355

and slide 'a' from 0 to 1.57 to 0, when 'b' is 0.785 or 2.355 , you'll see a square pyramid or tetrahedron morph into a cone.
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### Re: Implicit Definition of Other Shapes

Cool new 5D prism that has a sphere midsection. The pyramid to cylinder morph is neat. 5D Spherindrone IOOI>

I - Line segment: |x| - a = 0

IO - circle : √(x2 + y2) - a2 = 0

IOO - sphere : √(x2 + y2 + z2) - a2 = 0

IOOI - spherinder : |√(x2 + y2 + z2) - w| + |√(x2 + y2 + z2) + w| - a = 0

IOOI> - spherindrone : ||√(x2+y2+z2) -w| + |√(x2+y2+z2) +w| - 3v| + |√(x2+y2+z2) -w| + |√(x2+y2+z2) +w| - a = 0

abs(abs(sqrt(x^2+y^2+z^2) -w) + abs(sqrt(x^2+y^2+z^2) +w) - 3v) + abs(sqrt(x^2+y^2+z^2) -w) + abs(sqrt(x^2+y^2+z^2) +w) - a = 0

a = 8.5 , within XYZbox = -5,+5

3D Midsections of IOOI> along XYZWV

iiOI> : XY cut , II> sq pyramid
————————————————————————————————
abs(abs(sqrt(a^2+b^2+x^2) -y) + abs(sqrt(a^2+b^2+x^2) +y) - 3z) + abs(sqrt(a^2+b^2+x^2) -y) + abs(sqrt(a^2+b^2+x^2) +y) - 8.5
—— a : rounded collapse of pyramid height to line
—— b : rounded collapse of pyramid height to line

IOii> : ZW cut , IO> cone
——————————————————————————
abs(abs(sqrt(x^2+y^2+a^2) -b) + abs(sqrt(x^2+y^2+a^2) +b) - 3z) + abs(sqrt(x^2+y^2+a^2) -b) + abs(sqrt(x^2+y^2+a^2) +b) - 8.5
—— a : rounded collapse of cone height
—— b : flat collapse of cone height to circle

IOiIi : ZV cut , IOI cylinder
—————————————————————————————
abs(abs(sqrt(x^2+y^2+a^2) -z) + abs(sqrt(x^2+y^2+a^2) +z) - 3b) + abs(sqrt(x^2+y^2+a^2) -z) + abs(sqrt(x^2+y^2+a^2) +z) - 8.5
—— a : diameter collapse to line
—— b : mid-height taper symmetry , -b shrink, +b expand

IOOii : WV cut , IOO sphere
————————————————————————————
abs(abs(sqrt(x^2+y^2+z^2) -a) + abs(sqrt(x^2+y^2+z^2) +a) - 3b) + abs(sqrt(x^2+y^2+z^2) -a) + abs(sqrt(x^2+y^2+z^2) +a) - 8.5
—— a : linear symmetry, constant shape
—— b : mid-height taper symmetry , +b shrink, -b expand

IOOI> Rotation Function : YZ cut to WV cut, dual rotate
——————————————————————————————————————————————————————————
abs(abs(sqrt(x^2+(y*cos(a))^2+(z*cos(b))^2) -(y*sin(a))) + abs(sqrt(x^2+(y*cos(a))^2+(z*cos(b))^2) +(y*sin(a))) - 3(z*sin(b))) + abs(sqrt(x^2+(y*cos(a))^2+(z*cos(b))^2) -(y*sin(a))) + abs(sqrt(x^2+(y*cos(a))^2+(z*cos(b))^2) +(y*sin(a))) - 8.5

[a,b] - 3D Midsection Positions
——————
[0,0] - IOO sphere
[1.57,0] - IOI cylinder
[1.57,1.57] - II> square pyramid
[0,1.57] - IO> cone
Last edited by ICN5D on Sun Jan 04, 2015 5:46 pm, edited 1 time in total.
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### Re: Implicit Definition of Other Shapes

Just to keep track ...

• ...|     is described by     abs(... - an xn) + abs(... + an xn) = a0
• ...>     is described by     abs(... - an xn) + abs(...) = a0
• ...O     is described by     sqrt((...)2 + an2 xn2) = a0
where the "..." within the equations refers to the left part of the respective equation, corresponding to the previous symbol, etc.

Coefficients moreover have to be chosen appropriate to the demands, e.g.
• when spins should derive true circular shapes (i.e. not ellipses), then (I think) an ought to be equal to the circumradius of "...", right?
• And in order to have taperings of non-vanishing heights, then (I think) an ought to be chosen larger than the circumradius of "...", right?
Did I get it?

--- rk
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### Re: Implicit Definition of Other Shapes

From this algebraic description it should be somehow deducable
• that ||O (spinning square, i.e. cylinder) = |O| (extruded disc, i.e. cylinder again)
• and that |>O (spinning triangle, i.e. cone) = |O> (tapered disc, i.e. cone again)
But I cannot see that directly from their algebraic descriptions, even when assuming different coefficients.
(But that might be more my problem with wrangling algebraics right ... )

Might that be even a more general fact, i.e.
• would O and | generally commute?
• Resp. would O and > generally commute?
That should at least be (dis)provable by means of these algebraic forms.
(But at least | and > should not come out commutable in any case.)

--- rk
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### Re: Implicit Definition of Other Shapes

What would be the string for 6D cartesian product of two cones?

EDIT: From the symbols, I have a question about the spin O. Both prism and tapering operation are basically directionless, they can be applied only in one way. However, rotation requires a direction. A cylinder can be rotated in two different ways into 4D, for example: spherinder or duocylinder.

Toratopes (at least the basic ones) could be added into this system by a ring operator (R), which would signify a rotation around an external line/plane/hyperplane. The equation could be found by putting the previous shape shifted to -an and a reflected copy to +an and applying the spin operator to it.
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### Re: Implicit Definition of Other Shapes

Klitzing wrote:
• ...| is described by abs(... - an xn) + abs(... + an xn) = a0
• ...> is described by abs(... - an xn) + abs(...) = a0
• ...O is described by sqrt((...)2 + an2 xn2) = a0
where the "..." within the equations refers to the left part of the respective equation, corresponding to the previous symbol, etc.

Coefficients moreover have to be chosen appropriate to the demands, e.g.
• when spins should derive true circular shapes (i.e. not ellipses), then (I think) an ought to be equal to the circumradius of "...", right?
• And in order to have taperings of non-vanishing heights, then (I think) an ought to be chosen larger than the circumradius of "...", right?

Did I get it?

--- rk

Very close, the extrude is right, and most of the taper and rotate.

• [...]I : abs([...] - an xn) + abs([...] + an xn) = a0
• [...]> : abs([...] - an xn) + [...] = a0
• [...]O : is a bit more different. As Marek points out, we need to establish direction, and thus the stationary plane. This is highly dependent on the equation, since the notation is a bit ambiguous. This minor obstacle can be overcome with quickfur's suggestion of a subscript axis for the spin operator, Om. This will single out the moving axis 'm' , telling us what the stationary plane is. I'll detail this one further below.

To clarify, a0 is the circumradius for the entire overall shape. The value and usage of an with xn is only to scale the shape to a better proportion. Without it, we get rectangular prism for a cube, tall skinny pyramids/cones, etc. I found a nice relation to what value an should be. It has to do with how many repeated dimension terms there are, with respect to others.

• For example, the unit cube is:

abs(abs(x - y) + abs(x + y) - 2z) + abs(abs(x - y) + abs(x + y) + 2z) = a
There are two separate groups of X and Y, associated with only one Z. The circumradius 'a' is proportioned to those extra X's and Y's differently than Z. In order to get unit cube, we have to scale Z by a multiple of 2. This will balance out the Z extension, and make a perfect cube.

• However, with a unit cylinder:

abs(sqrt(x^2 + y^2) - z) + abs(sqrt(x^2 + y^2) + z) = a
The X, Y, and Z dimensions are proportioned equally, needing no scaling to get unit cylinder.

• For tapered pyramids, the right triangle :

abs(abs(x) - 2y) + abs(x) = a
Circumradius 'a' is proportioned to 2 X's, and only one Y. Scaling Y by 2-fold will make a nicer looking triangle. I guess it's really only cosmetic outside the usage with a unit cube.

• For square pyramid,

abs(abs(x - y) + abs(x + y) -3z) + abs(x - y) + abs(x + y) = a
I found scaling Z by 3 instead of 2 makes a better looking shape. Again, this is more of a cosmetic thing.

Klitzing wrote:From this algebraic description it should be somehow deducable
• that ||O (spinning square, i.e. cylinder) = |O| (extruded disc, i.e. cylinder again)
• and that |>O (spinning triangle, i.e. cone) = |O> (tapered disc, i.e. cone again)
But I cannot see that directly from their algebraic descriptions, even when assuming different coefficients.
(But that might be more my problem with wrangling algebraics right ... )

Might that be even a more general fact, i.e.
• would O and | generally commute?
• Resp. would O and > generally commute?
That should at least be (dis)provable by means of these algebraic forms.
(But at least | and > should not come out commutable in any case.)

--- rk

Oh, yes! The commutative operators show themselves very nicely in the algebraic definition. But, they can commute only after some combinations of extrude or taper operators. They can be a little tricky. And, of course, the taper + extrude does not commute , [...]I> =/= [...]>I

• IOI = IIO
• IO> = I>O
• IIIO = IIOI = IOII
• IIO> = IOI> = II>O
• IO>I = I>OI =/= IOI> =/= I>IO

I will explain the bisecting rotate here as well. Simply put, a bisecting rotation will single out an axis to become the rotating axis into the higher dimension. The axis, which cuts as a line segment (through the shape), becomes a circular parameter. If rotating an n-sphere parameter, we simply fatten it up into an n+1 sphere.

So, for the IO and OI relationship:

1) Starting with line segment, I

|x| = a
2) Bisecting Rotate into 2D, around stationary 0-plane makes circle, IO. Since no axis is stationary, the moving axis will be X. Line segment |x| becomes "circle-segment" sqrt(x^2 + y^2).

sqrt(x^2 + y^2) = a
3) Extrude into 3D, making cylinder IOI. Extrusions are the simplest cartesian product, where [...]I is the ([...] , line) prism, as infinite shapes [...] stacked with a line segment.

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

Conversely, the second way to build a cylinder would be to interchange the spin and extrude operators:

|x| = a
2) Extrude into 2D, making square, as infinite lines stacked within line segment, II

|x - y| + |x + y| = a
3) Bisecting rotate square into 3D, around stationary 1-plane Y. Either X or Y will work. This will turn x into sqrt(x^2 + z^2), within the square equation:

|sqrt(x^2 + z^2) - y| + |sqrt(x^2 + z^2) + y| = a
which is of course equal to the other orientation

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

Now for cone, and the commuting O> and >O :

|x| = a
2) Taper to point along 2D, making right triangle I> :

||x| - 2y| + |x| = a
3) Bisecting rotate into 3D, around stationary 1-plane Y. If using stationary 1-plane X, we get something like a self-intersecting triangle torus, with no major diameter. I call these strange entities cyclotopes, as they are artifacts of the bisecting rotation. Again, as with cylinder, we turn |x| into sqrt(x^2 + z^2):

|sqrt(x^2 + z^2) - 2y| + sqrt(x^2 + z^2) = a
which is equal to the other orientation

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

And, the other way to build cone:

1) Starting with line segment, I

|x| = a
2) Bisecting Rotate into 2D, around stationary 0-plane makes circle, IO. Line segment |x| becomes "circle-segment" sqrt(x^2 + y^2).

sqrt(x^2 + y^2) = a
3) Taper the circle to point along 3D, making cone IO>. This will use the standard taper transformation along axis n, [...]>n : |[...] - xn| + [...] = a

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

For rotating cylinder into 4D, we use the same approach as replacing xn with sqrt(xn^2 + xn+1^2). If the moving axis is in an n-sphere parameter, we simply add another dimension within, and fatten to an N+1 sphere.

Cylinder IOI or IIO :

|sqrt(x^2 + y^2) - z| + |sqrt(x^2 + y^2) + z| = a
1) Bisecting rotate into W, around stationary 2-plane XY, will leave Z as axis in motion. X and Y will remain unchanged, but Z will become sqrt(z^2 + w^2), making duocylinder IOIO, infinite circles stacked within circle-segment:

|sqrt(x^2 + y^2) - sqrt(z^2 + w^2)| + |sqrt(x^2 + y^2) + sqrt(z^2 + w^2)| = a
2) Bisecting rotate into W, around stationary 2-plane XZ or YZ, will leave Y or X as the axis in motion, respectively. Since X or Y are already inside a circle parameter, sqrt(x^2 + y^2) becomes sqrt(x^2 + y^2 + w^2), making the spherinder IOOI :

|sqrt(x^2 + y^2 + w^2) - z| + |sqrt(x^2 + y^2 + w^2) + z| = a
which is another orientation of

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

Marek14 wrote:What would be the string for 6D cartesian product of two cones?

That one would be IO>[IO>] , the duoconinder. The commuting O> within the cones will hold true, as well. Defined algebraically, as infinite cones XYZ stacked within a 'cone-segment' WVU:

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

Marek14 wrote:Toratopes (at least the basic ones) could be added into this system by a ring operator (R), which would signify a rotation around an external line/plane/hyperplane. The equation could be found by putting the previous shape shifted to -an and a reflected copy to +an and applying the spin operator to it.

Yes, that's my fiber bundle operator, A(B) , shape A stretched over surface of shape B. It's a modification of the bisecting rotate, since it acts on only one axis, leaving the others alone. But, instead of turning an axis into a circle-segment, we replace with the whole circle (or whichever shape) along with its circumradius. So, for triangle torus I>(O),

||x| - 2y| + |x| = a
non-bisecting rotate around 1-plane Y, into Z, will turn |x| into |(sqrt(x^2 + z^2) - b)| . The absolute values have to be present around the hollow circle parameter for non-bisecting rotate, if originally there for the transformed axis.

||(sqrt(x^2 + z^2) - b)| - 2y| + |(sqrt(x^2 + z^2) - b)| = a
and non-bisecting rotate around 1-plane X, into Z, will turn 2y into 2(sqrt(y^2 + z^2) - b)

||x| - 2(sqrt(y^2 + z^2) - b)| + |x| = a
as the two orientations of a triangle torus I>(O).

For toratopes, the spin operators are within the (), as in (O) for over a circle, (OO) for over sphere, (O)(O) as over a torus, etc. The (O) symbol is the standard non-bisecting rotate, but can be expanded for the surface of other shapes. The symbol (>) is the 1-surface of triangle I>, and (I) is the 1-surface of square II. Simply remove the first extrusion, and place all remaining symbols within the () operator, to denote just the surface of the shape.

I use [(O)(O)] for the duocylinder margin for now, as the product of the 1-surface of two circles. Logically, it may have a better symbol, as in ([O][O]) , or (O)[(O)] , etc. For the 3-manifolds, I use (O)[(O)(O)] for surface of tiger, (O)(O)(O) for surface of ditorus, [(O)(O)](O) for cyltorinder (torus of duocylinder) margin, [(OO)(O)] for 3-frame of cylspherinder IOOIO , (OOO) for surface of glome, etc. I define the cyltrianglintigroid as IO[(>)(O)] , circle over the 2-frame of cyltrianglinder I>IO.

These operators do not commute for closed toratopes, but only a certain way for opens. For example, IO(O)IO = IOIO(O) , cyltorinder is equal to duocylinder torus, since IOIO = IO[IO] . The whole circle product commutes and cannot be split up.

IOO / (III) - SPHERE
IO(O) / ((II)I) - TORUS

IO[(O)(O)] / ((II)(II)) - TIGER
IO(O)(O) / (((II)I)I) - DITORUS
IO(OO) / ((III)I) - TORISPHERE
IOO(O) / ((II)II) - SPHERITORUS
IOOO / (IIII) - GLOME

IOOOO / (IIIII) - PENTASPHERE
IOOO(O) / ((II)III) - GLOMITORUS
IOO[(O)(O)] / ((II)(II)I) - SPHERITIGER
IOO(OO) / ((III)II) - SPHERISPHERE
IOO(O)(O) / (((II)I)II) -SPHERIDITORUS
IO[(OO)(O)] / ((III)(II)) - CYLSPHERINTIGROID
IO[(O)(O)](O) / (((II)I)(II)) - TIGER TORUS
IO(OOO) / ((IIII)I) - TORIGLOME
IO(OO)(O) / (((II)II)I) - TORISPHERITORUS
IO(O)[(O)(O)] / (((II)(II))I) - TORATIGER
IO(O)(OO) / (((III)I)I) - DITORISPHERE
IO(O)(O)(O) / ((((II)I)I)I) - TRITORUS

It gets a bit more ambiguous in 8D, with shapes like (((III)II)((II)I)) and (((II)II)((III)I)) being equal to my symbol IO[(OO)(O)][(OO)(O)] . There would need to be some modification to denote the difference.
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### Re: Implicit Definition of Other Shapes

I finally got around to exploring the tesseract, which offered more than I expected for cool topology morphs. Awesome to see the implicit surface equation works, too!

Implicit Definition of Tesseract:

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

abs(abs(x-y)+abs(x+y) - abs(z-w)-abs(z+w)) + abs(abs(x-y)+abs(x+y) + abs(z-w)+abs(z+w)) = a

It's the cartesian product of two orthogonal squares, XY and ZW, very simple.

In order to turn the 4-cube so that it slides corner first, I needed to set up a triple-nested rotate+translate function, with 3 startpoints and one endpoint.

One axis rotate: X to A
x -> (x*sin(b) + a*cos(b))
a -> (x*cos(b) - a*sin(b))

Two-Axis Rotate: X to A , Y to A
x -> (x*sin(b) + a*cos(b))
y -> (y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))
a -> (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))

Three-Axis Rotate: X to A , Y to A , Z to A
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))
a —> (z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d))

Triple-Axis Rotate of Tesseract
—————————————————
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))) - abs((z*sin(d) + (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*cos(d))-(z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d)))-abs((z*sin(d) + (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*cos(d))+(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))) + abs((z*sin(d) + (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*cos(d))-(z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d)))+abs((z*sin(d) + (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*cos(d))+(z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d)))) = 10

-5 < a < 5
0 < b,c,d < 1.57
XYZbox = -5 / +5

At a=0 midsection, perfect octahedron at angles [b,c,d] = [1.04667,0.96737,0.80879] . Use for sliding through corner first.

At a=-1.1 : truncated tetrahedron

Tesseract passing through corner first: Set [b,c,d] to [1.04667,0.96737,0.80879] , then slide 'a' from -5 to +5 for corner-first. Neat to see how the expanding tetrahedron begins to truncate, and morph into an octahedron at perfect midsection.

Tesseract passing 1D edge first: Set [b,c,d] to [1.57,0.96737,0.80879], then slide 'a' from -5 to +5 for 1D edge first. The triangle prism was unexpected to me. I didn't think it would make that, but it makes sense when thinking about it. When sliding a 3D cube corner first, we get a triangle. A tesseract is made by extending the cube into 4d. Which means, the triangle-slice pattern from the cube gets extended as well, into a triangle prism.

Tesseract passing square-face first: Set [b,c,d] to [1.57,1.57,0.80879], then slide 'a' from -5 to +5 for 2D square-face first. This one is closely related to passing a cube through 2D line-edge first. We get a line that expands into square, contracts to line. A hypercube makes an extended version of that, again.

Tesseract Rotation : A very cool morph of cube to octahedron to hexagon prism to octahedron to cube. HQ VERSION : https://gfycat.com/UnsightlyDefensiveBurro

Octahedron midsection: Hexagon Prism midsection: Truncated tetrahedron: in search of combinatorial objects of finite extent
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### Re: Implicit Definition of Other Shapes

Yes, I tried out sections of various uniform polytopes in Stella, so I'm familiar with these ones Marek14
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### Re: Implicit Definition of Other Shapes

It's awesome, isn't it Taking that hypothetical equation, and manipulating the shape, then verifying all the sections is most rewarding. It's like toratope exploration all over again.
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### Re: Implicit Definition of Other Shapes

The cross-sections of the tesseract are well-known, for example, from a page on my website: This must be the first time I've seen an animation of the cross-sections of a rotating tesseract, though. It looks really great. Fun stuff!!
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### Re: Implicit Definition of Other Shapes

Here's the 5D list of implicit definitions, including toratopes:

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(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) - v| + |sqrt(x²+y²+z²+w²) + v| - 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+y²z²) + 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+y²z²) + |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: ||||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)| +4v| - a

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
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### Re: Implicit Definition of Other Shapes

It's funny, because I was just talking about this shape. I happen to come across the algebraic definition of tegums, and write the implicit equation for a bicircular tegum. It's a modified version of a pyramid operation.

Square pyramid
||x-y|+|x+y| + 3z| + |x-y|+|x+y| = a

Octahedron: <square,line>
||x-y|+|x+y| + 2|z|| + |x-y|+|x+y| = a

cone
|√(x² + y²) + 2z| + √(x² + y²) = a

Bicone : <circle,line>
|√(x² + y²) + 2|z|| + √(x² + y²) = a

General pattern:

<m,n> tegum == |m + 2n| + m

the shape/equation for n must be defined as full line segment |xn| , for <m,line> tegums

Now the real test: take the <circle,line> bicone, and try replacing the line segment |z| with a circle segment √(z² + w²) , making hypothetical <circle,circle> bicircular tegum equation:

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

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

The cuts of this function has two orthogonal bicones, with a 45 degree oblique midsection of a pillow with four points in a square

X-cut : Bicone <circle,line> tegum
abs(sqrt(a^2 + y^2) + 2sqrt(z^2 + w^2)) + sqrt(a^2 + y^2) = 10

W-cut : Bicone , <circle,line> tegum
abs(sqrt(x^2 + y^2) + 2sqrt(z^2 + a^2)) + sqrt(x^2 + y^2) = 10

X -> W rotation , morph between two orthogonal <circle,line> tegums
abs(sqrt((x*sin(a))^2 + y^2) + 2sqrt(z^2 + (x*cos(a))^2)) + sqrt((x*sin(a))^2 + y^2) = 10

X -> W rotate + translate , full control A slide, B rotate
abs(sqrt((x*sin(b) + a*cos(b))^2 + y^2) + 2sqrt(z^2 + (x*cos(b) - a*sin(b))^2)) + sqrt((x*sin(b) + a*cos(b))^2 + y^2) = 10

Using the rotate + translate function, here are some animations:

Bicircular Tegum Passing through a 3-plane Bicircular Tegum 90 degree Rotation X -> W Bicircular Tegum 360 degree rotate, held at depth of a=2 from center Bicircular Tegum passing through a 3-plane at 5 angles of rotation X -> W Can these midsections be verified? Has anyone seen bicircular tegum before that can validate them?

EDIT : spelling
Last edited by ICN5D on Wed Jan 28, 2015 2:13 am, edited 1 time in total.
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### Re: Implicit Definition of Other Shapes

Wow, the second-from-last animation looks like someone kidnapped a triangle, put it into a pillowcase, and now it's trying to fight its way out...
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### Re: Implicit Definition of Other Shapes

Marek14 wrote:Wow, the second-from-last animation looks like someone kidnapped a triangle, put it into a pillowcase, and now it's trying to fight its way out...

LOL!!! quickfur
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### Re: Implicit Definition of Other Shapes

Yeah it does! That one has an eye-catching and aggressive topology morph, had to make it.
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### Re: Implicit Definition of Other Shapes

You should really get your own wiki section... You could have links to all shapes you've explored there, with equations and gifs. Putting it here will become hard to track in a while.
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### Re: Implicit Definition of Other Shapes

How complete are the 4D and 5D lists, btw? I noticed that you, for example, define I>(O), but not II(O) (square stretched over circle) or IO(II) (vice versa).

With tegums, the only wrinkle is that there are now two distinct kinds of squares -- prism type [line,line] and tegum type <line,line>, which must be distinguished from each other because they can lead to different shapes. Let's call the tegum square <II> a "rhombus". Similarly, higher-D shapes that use squares in their definitions will now also get more forms.

The good news is that if you include tegums, you basically don't need the taper operator anymore. Any X> is basically a half of <X,line> (topologically speaking), meaning that you can discuss it together with the tegum. This also allows you to have ALL coordinate mid-cuts nicely defined (tapers are either all in one half-space or their cuts get weird).

Using this, I'll post how I see the shapes in small dimensions in a bracket notation that is a bit different from yours:

0D
. - point

1D
| - line

2D
[II] - square
(II) - circle
<II> - rhombus - identical to square; triangle I> is one half of rhombus.

3D
[III] - cube
([II]I) - dome, crind
<[II]I> - square dipyramid - identical to octahedron; half is square pyramid
[(II)I] - cylinder
(III) - sphere
<(II)I> - bicone; half is cone
[<II>I] - rhombus prism - identical to cube; half is triangular prism
(<II>I) - rhombus dome - identical to dome; half is some strange shape, apparently
<III> - octahedron
Toratopes:
[II]-[II] - square/square torus
[II]-(II) - square/circle torus
[II]-<II> - square/rhombus torus - identical to square/square torus; half can be modified into square/triangle torus
(II)-[II] - circle/square torus
(II)-(II) - torus
(II)-<II> - circle/rhombus torus - identical to circle/square torus; half can be modified into circle/triangle torus
<II>-[II] - rhombus/square torus; half is triangle/square torus
<II>-(II) - rhombus/circle torus; half is triangle/circle torus
<II>-<II> - rhombus/rhombus torus - identical to rhombus/square torus; half is triangle/rhombus torus; another half can be modified into rhombus/triangle torus; quarter can be modified into triangle/triangle torus

Generally, this would encompass graphotopes (although the notation is insufficient for all of them), and also their expanded form, matrixtopes which are defined by n x n symmetrical matrix where every nondiagonal element defines whether the planar mid-cut by these two dimensions is a square, circle or rhombus. It would also use the fiber bundle operator.

Let's ask: How would the square tiger and rhombus tiger look?
Square tiger would be [II]-[(II)(II)], square stretched over duocylinder, while rhombus tiger would be analogical <II>-[(II)(II)].
First of all, the fiber bundles have more dimensional markers than the actual dimension. That is because the shape stretched over the base has two kinds of dimension: internal, which are already present in the base, and external. We can distinguish this by putting the internal dimensions in lowercase:
[ii]-[(II)(II)]
This is still probably not enough, with more complicated shapes there might be multiple ways to fit them over the base.

So, how would the cuts of square tiger look like?

[ii]-[(II)(I)]

This represents square stretched over product of circle (II) and (I), which is two points, i.e. vertical stack of two square/circle toruses. All cuts look like this. Similar for rhombus tiger, whose cuts are all vertical stacks of rhombus/circle toruses. Note that [I], (I) and <I> are ALL pair of points or line, they only differ in how the midcut changes when moving in that dimension.

Similarly, you can have a circle stretched over the surface of cube (Ii)-[III] with three cuts looking like circle/square torus and one looking like two concentric cubes.

Circle/square torus itself (Ii)-[II] will have two cuts looking like two circles and third looking like two concentric squares.

Note that circle/rhombus and circle/square toruses wouldn't actually look exactly the same, as one would have squares in the corner-to-corner cuts while the other would probably have ellipses there, but they are the same category of shapes.
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### Re: Implicit Definition of Other Shapes

Let's have a look at simple 3D matrixtopes using all three kinds.

The following types exist (using symbols S, C and R):

3 3-types:
SSS - cube [III], all cuts squares
CCC - sphere (III), all cuts circles
RRR - octahedron <III>, all cuts rhombi

6 21-types:
SSC - cylinder [(II)I], one cut circle, two cuts squares
SSR - rhombus prism [<II>I], isomorphic to cube, one cut rhombus, two cuts squares
CCS - dome ([II]I), one cut square, two cuts circles
CCR - rhombic dome (<II>I), isomorphic to dome, one cut rhombus, two cuts circles
RRS - dipyramid <[II]I>, isomoephic to octahedron, one cut square, two cuts rhombi
RRC - bicone <(II)I>, one cut circle, two cuts rhombi

1 111-type:
SCR - trinity, one cut square, one cut circle, one cut rhombus -- this is a mixed matrixtope that can't be expressed in bracket notation.
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### Re: Implicit Definition of Other Shapes

In 4D, then, the matrixtopes would be (ordering xy, xz, xw, yz, yw, zw):

3 6-types:
SSSSSS - tesseract [IIII], all cuts cubes
CCCCCC - glome (IIII), all cuts spheres
RRRRRR - 16-cell <IIII>, all cuts octahedra

6 51-types:
SSSSSC - cubinder [(II)II], two cuts cylinders, two cuts cubes
SSSSSR - square/rhombus duoprism [<II>II], isomorphic to tesseract, two cuts rhombic prisms, two cuts cubes
CCCCCS - semiglome ([II]II), two cuts domes, two cuts spheres
CCCCCR - rhombic semiglome (<II>II), isomorphic to semiglome, two cuts rhombic domes, two cuts spheres
RRRRRS - square/rhombus tegum <[II]II>, isomorphic to 16-cell, two cuts dipyramids, two cuts octahedra
RRRRRC - circle/rhombus tegum <(II)II>, two cuts bicones, two cuts octahedra

6 42a-types:
SSSSCC - dominder [([II]I)I], one cut dome, one cut cube, two cuts cylinders
SSSSRR - dipyramid prism [<[II]I>I], isomorphic to octahedral prism, one cut dipyramid, one cut cube, two cuts rhombic prisms
CCCCSS - spheridome ([(II)I]I), one cut cylinder, one cut sphere, two cuts domes
CCCCRR - ??? (<(II)I>I), one cut bicone, one cut sphere, two cuts rhombic domes
RRRRSS - rhombic prism dipyramid <[<II>I]I>, isomorphic to cube dipyramid, one cut rhombic prism, one cut octahedron, two cuts dipyramids
RRRRCC - rhombic dome tegum <(<II>I)I>, one cut rhombic dome, one cut octahedron, two cuts bicones

6 42b-types:
CSSSSC - duocylinder [(II)(II)], all cuts cylinders
RSSSSR - rhombus duoprism [<II><II>], isomorphic to tesseract, all cuts rhombic prisms
SCCCCS - cyclodome ([II][II]), all cuts domes
RCCCCR - rhombic cyclodome (<II><II>), isomorphic to cyclodome, all cuts rhombic domes
SRRRRS - square/square tegum <[II][II]>, isomorphic to 16-cell, all cuts dipyramids
CRRRRC - duocone <(II)(II)>, all cuts bicones

3 411a-types:
SSSSCR - trinity prism, one cut trinity, one cut rhombic prism, one cut cylinder, one cut cube
CCCCSR - trinisphere, one cut trinity, one cut rhombic dome, one cut dome, one cut sphere
RRRRSC - trinity dipyramid, one cut trinity, one cut dipyramid, one cut bicone, one cut octahedron

3 411b-types:
CSSSSR - circle/rhombus duoprism [(II)<II>], isomorphic to cubinder, two cuts cylinders, two cuts rhombic prisms
SCCCCR - semirhombic cyclodome ([II]<II>), isomorphic to cyclodome, two cuts domes, two cuts rhombic domes
SRRRRC - square/circle tegum <[II](II)>, isomorphic to circle/rhombus tegum, two cuts dipyramids, two cuts bicones

3 33a-types:
SSCSCC - longdome, two cuts domes, two cuts cylinders
SSRSRR - ???, two cuts dipyramids, two cuts rhombic prisms
CCRCRR - ???, two cuts bicones, two cuts rhombic domes

6 33b-types:
SSSCCC - spherinder [(III)I], one cut sphere, three cuts cylinders
SSSRRR - octahedral prism [<III>I], one cut octahedron, three cuts rhombic prisms
CCCSSS - tridome ([III]I), one cut cube, three cuts domes
CCCRRR - ??? (<III>I), one cut octahedron, three cuts rhombic domes
RRRSSS - cube dipyramid <[III]I>, one cut cube, three cuts dipyramids
RRRCCC - bisphericone <(III)I>, one cut sphere, three cuts bicones

To be continued...
Last edited by Marek14 on Tue Jan 27, 2015 9:03 pm, edited 1 time in total.
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### Re: Implicit Definition of Other Shapes

6 321a-types:
SSCCSR - two cuts trinity, two cuts cylinder
SSRRSC - two cuts trinity, two cuts rhombic prism
CCSSCR - two cuts trinity, two cuts dome
CCRRCS - two cuts trinity, two cuts rhombic dome
RRSRRC - two cuts trinity, two cuts dipyramid
RRCCRS - two cuts trinity, two cuts bicone

6 321b-types:
SSCRSC - one cut trinity, one cut dome, one cut cylinder, one cut rhombic prism
SSRCSR - one cut trinity, one cut dipyramid, one cut rhombic prism, one cut cylinder
CCSRCS - one cut trinity, one cut cylinder, one cut dome, one cut rhombic dome
CCRSCR - one cut trinity, one cut bicone, one cut rhombic dome, one cut dome
RRSCRS - one cut trinity, one cut rhombic prism, one cut dipyramid, one cut bicone
RRCSRC - one cut trinity, one cut rhombic dome, one cut bicone, one cut dipyramid

6 321c-types:
SSSCCR - rhombic dome prism [(<II>I)I], isomorphic to dominder, one cut rhombic dome, one cut rhombic prism, two cuts cylinder
SSSRRC - bicone prism [<(II)I>I], one cut bicone, one cut cylinder, two cuts rhombic prism
CCCSSR - rhombic tridome ([<II>I]I), isomorphic to tridome, one cut rhombic prism, one cut rhombic dome, two cuts dome
CCCRRS - ??? (<[II]I>I), isomorphic to (<III>I), one cut dipyramid, one cut dome, two cuts rhombic dome
RRRSSC - dome tegum <([II]I)I>, isomorphic to rhombic dome tegum, one cut dome, one cut dipyramid, two cuts bicone
RRRCCS - cylinder tegum <[(II)I]I>, one cut cylinder, one cut bicone, two cuts dipyramid

2 222a-types:
SCCSRR - one cut dipyramid, one cut rhombic dome, one cut trinity, one cut cylinder
SRRSCC - one cut dome, one cut bicone, one cut trinity, one cut rhombic prism

3 222b-types:
SSCCRR - one cut bicone, two cuts trinities, one cut cylinder
SSRRCC - one cut rhombic dome, two cuts trinities, one cut rhombic prism
CCSSRR - one cut dipyramid, two cuts trinities, one cut dome

1 222c-type:
RSCCSR - all cuts trinities
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### Re: Implicit Definition of Other Shapes

Now, for 4D fiber bundles, there are some fundamental types based on dimensions:

2D stretched over 3D (1 internal dimension) like torisphere -- should work for any combination

2D stretched over 4D (2 internal dimensions) like tiger -- needs the base to have connected two-dimensional skelet. So circle-glome (ii)-(IIII) doesn't work because glome has no elements of dimension 2, but circle-duocylinder, i.e. tiger (ii)-[(II)(II)] works because duocylinder has a 2D element, the cartesian product of the two circles. And circle-spherinder (ii)-[(III)I] doesn't work because spherinder has two sphere surfaces as elements of dimension 2, but they are not connected together, meaning that this would just be stack of two torispheres -- can occur as a cut of higher-D shape, but it's not a separate shape.
But from this it SHOULD be possible to do things like circle-tesseract (ii)[IIII], which would add circles to every point of 2D tesseract skelet.

3D stretched over 2D (1 internal dimension) like spheritorus or ditorus -- should work for any combination

3D stretched over 3D (2 internal dimensions) and 3D stretched over 4D (3 internal dimensions) -- work if there are connected 1D skelets. So you can define sphere-cube and even sphere-tesseract, just as you can imagine circle-cube in 3D.
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### Re: Implicit Definition of Other Shapes

You should really get your own wiki section...

Yes, I know. I've been lacking in that dept for a while. Or ever, is more like it. There's quite a bit of toratope stuff alone!

How complete are the 4D and 5D lists, btw? I noticed that you, for example, define I>(O), but not II(O) (square stretched over circle) or IO(II) (vice versa).

I experimented around with those, and just haven't typed the list yet. Explored a rotate function of II>(O), but not much else. And, after defining cyltrianglintigroid, the method was made clear with inflating a triangle frame. I've also wondered about inflating just the 1D edge of a 3D shape, with a circle. Imagine a true circular wireframe of a square pyramid. I have no idea how to do that ..... yet. Certainly we can cheat by using four IO(>) circle,triangle tori plus a IO(I) circle,square torus, expressed as a product of all five in the necessary arrangement. It would be some sort of strange class of 3D tigroid-like shapes, with 4D variations, as well.

The good news is that if you include tegums, you basically don't need the taper operator anymore. Any X> is basically a half of <X,line> (topologically speaking), meaning that you can discuss it together with the tegum. This also allows you to have ALL coordinate mid-cuts nicely defined (tapers are either all in one half-space or their cuts get weird).

Yes, this is true. Maybe I'll add it without ditching the taper just yet, and expand my list with the inclusion. Only because I'm not sure what else to call a half-tegum operation other than a taper. A hemi-pyramid? Hemi-tegum? Plus, I do have a concise algebraic means to just taper something.

But from this it SHOULD be possible to do things like circle-tesseract (ii)[IIII], which would add circles to every point of 2D tesseract skelet.

That right there looks like a very interesting experiment. I wonder..... it could make a circle over 1-frame of cube, a wireframe cube with discontinuity at both ends in 4D. Wow, might have just found what I was looking for, in above ponderings. Need to try!

Using this, I'll post how I see the shapes in small dimensions in a bracket notation that is a bit different from yours:

In which case is an excellent notation. There are way more possibilities than I imagined. It seems like the matrixtopes came first, then a more abstract way to combine them with known types, becoming bracketopes. There's quite a bit of good stuff in here, many of which I can render now. Thanks!
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### Re: Implicit Definition of Other Shapes

Note, however, that there are various inequivalent expansion possibilities. For example, the 2-frame of tesseract will look different when considering it as cartesian product of two squares (then it will be product of their circumferences, 16 squares in total, topological analogue of duocylinder margin), when you consider it as cube prism (then it will form two separated groups of 6 squares) and when you consider tesseract as fully symmetrical shape (then it will have full 24 squares). Connected 1-frame of tesseract is only possible when you consider it as elementary shape.
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### Re: Implicit Definition of Other Shapes

Circle over 2-frame of Tesseract Experiments

Tesseract equation:

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

abs(abs(x-y)+abs(x+y) - abs(z-w)-abs(z+w)) + abs(abs(x-y)+abs(x+y) + abs(z-w)+abs(z+w)) = a

• Attempting to assign as product of 1-surface of two ortho squares, then expand with circle

The Tesseractigroid IO[(I)(I)]

(√(|x-y|+|x+y|) - a)² + (√(|z-w|+|z+w|) - b)² = c²

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

• 3D cuts

Set b,c = 2.5 / d = 0.5

W-cut
(sqrt(abs(x-y)+abs(x+y)) -b)^2 + (sqrt(abs(z-a)+abs(z+a)) -c)^2 = d^2

Makes stack of IO(I), with discont at 4D ends

Y -> W rotate
(sqrt(|x-(y*sin(a))|+|x+(y*sin(a))|) -b)^2 + (sqrt(|z-(y*cos(a))|+|z+(y*cos(a))|) -c)^2 = d^2

Oblique midsection gives the Cage of Tesseractigroid This is as close as I got. I tried many other experiments, but no success of making full circle over 1-frame of cube. Circle over 2-frame of tesseract needs that more general, elementary equation. You provided an equation for hexagon vertices : (x + iy)^6 , so is there an analogue for lines, squares, etc, of higher dimensional shapes? Would they use complex numbers as well? Seems like an interesting problem, in general.
Last edited by ICN5D on Thu Jan 29, 2015 6:09 pm, edited 1 time in total.
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### Re: Implicit Definition of Other Shapes

My guess about circle over 1-frame of cube is that this failed because you don't have fully symmetrical expression for the 1-frame. When expressing cube as [[II]I], the 1-frame will be two squares.

However, what happens if you use circle-octahedron? The octahedron expressed as <<II>I> should have complete 1-frame.
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### Re: Implicit Definition of Other Shapes

circle over octahedron experiment

|x| + |y| + |z| = a

(sqrt(|x| + |y| + |z|) - a)^2 = b^2
- smaller oct within larger (sqrt(|x|) -a)^2 + (sqrt(|y|) -a)^2 + (sqrt(|z|) -a)^2 = b^2
- blobs in triger cut ((I)(I)(I))

other octahedron equation:

||x-y|+|x+y| + 3|z|| + |x-y|+|x+y| = a

abs(abs(x-y)+abs(x+y) + 3abs(z)) + abs(x-y)+abs(x+y) = a

|(√(|x-y|)-a)^2 + (√(|x+y|)-a)^2 + 3|z|| + (√(|x-y|)-a)^2 + (√(|x+y|)-a)^2 = b^2

abs((sqrt(abs(x-y))-a)^2 + (sqrt(abs(x+y))-a)^2 + 3abs(z)) + (sqrt(abs(x-y))-a)^2 + (sqrt(abs(x+y))-a)^2 = b^2
- morphs an octahedron into four blobs in 2x2 square array

|(√(|x-y|)-a)^2 + (√(|x+y|)-a)^2 + 3(√(|z|)-a)^2| + (√(|x-y|)-a)^2 + (√(|x+y|)-a)^2 = b^2

abs((sqrt(abs(x-y))-a)^2 + (sqrt(abs(x+y))-a)^2 + 3(sqrt(abs(z))-a)^2) + (sqrt(abs(x-y))-a)^2 + (sqrt(abs(x+y))-a)^2 = ((-3.2a/1.8 )+4.2)^2
- morphs an octahedron to a 2x2x2 array of 8 blobs , set 0 < a < 1.9 --->> check this one out, really cool (√(||x-y|+|x+y| + 3|z||) -a)^2 + (√(|x-y|+|x+y|) -a)^2 = b^2

(sqrt(abs(abs(x-y)+abs(x+y) + 3abs(z)))-a)^2 + (sqrt(abs(x-y)+abs(x+y))-a)^2 = (((-4.2a/3)+4.25)^2)
- Adjusting ‘a’ morphs octahedron into circle over square, set 0 < a < 2.75 --->> and this one

No luck getting anything close. Some strange morphs, though.
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### Re: Implicit Definition of Other Shapes

It partially reminds me of blastocyst.

The first image is basically line over octahedron. When extended to 4D, it would be circle over 2D surface of octahedron (Ii)-<III>, a variant of torisphere, with three cuts (Ii)-<II> (circle over rhombus) and fourth the two octahedra from your first image.

Circle over 1-frame of octahedron would be (ii)-<III>, would be a 3D shape, with 4D analogue (Iii)-<III>.

What I'm interested in at this point is the shape I called "trinity"...

max(x^2,y^2)=1
y^2 + z^2 = 1
|x| + |z| = 1

max(x^2,y^2 + z^2,(|x| + |z|)^2) = 1

x = 0 cut:
max(0,y^2 + z^2,(0 + |z|)^2) = 1
max(0,y^2 + z^2,(|z|)^2) = 1
max(0,y^2 + z^2,z^2) = 1
y^2 + z^2 >= 0, y^2 + z^2 >= z^2
y^2 + z^2 = 1

y = 0 cut:
max(x^2,0 + z^2,(|x| + |z|)^2) = 1
max(x^2, z^2,(|x| + |z|)^2) = 1
(|x| + |z|)^2 >= x^2, (|x| + |z|)^2 >= z^2
(|x| + |z|)^2 = 1
|x| + |z| = 1

z = 0 cut:
max(x^2,y^2 + 0,(|x| + 0)^2) = 1
max(x^2,y^2,(|x|)^2) = 1
max(x^2,y^2,x^2) = 1
max(x^2,y^2) = 1

So it seems to work. Can you have a look at this creature? Marek14
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### Re: Implicit Definition of Other Shapes

Using the three term max function, I get the equation

((x^2 + y^2 + z^2 + abs(x^2 - y^2 - z^2))/2 + (|x| + |z|)^2 + abs((x^2 + y^2 + z^2 + abs(x^2 - y^2 - z^2))/2 - (|x| + |z|)^2))/2 = a

which makes

Seen at angle Circle Midsection Angle Rhombus Midsection Angle Square Midsection Angle wow, really cool! I see how that works, now. The 4D varieties make a little more sense after seeing a 3D version.
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### Re: Implicit Definition of Other Shapes

Yes, glad that works Actually, now I realize that you don't actually need the 3-maximum function: since (|x| + |z|)^2 >= x^2, x^2 can be omitted to a simpler equation max(y^2 + z^2,(|x| + |z|)^2) = 1.

Which would mean that it's pretty simple to create all kinds of functions for other matrixtopes. The weirdest one is probably the RSCCSR, which has trinities as all four cuts, and whose equation should be:

max((|x| + |y|)^2, x^2 + w^2, y^2 + z^2, (|z| + |w|)^2) = 1
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