## Toratope Names

Discussion of shapes with curves and holes in various dimensions.

### Toratope Names

Hi everyone,
I'm a little confused about the naming conventions of the toratopes. In 4D we have the cubinder which is short for cubic cylinder, which assumedly means a lathing of the cube (and a 3D cylinder would really be in full a "squaric cylinder"). But then what is the term spherinder? Both the duocylinder and spherinder are lathings of the 3D cylinder (in 2 different directions). A lathing of the sphere would be a glome. Also the torinder is not a lathing of the torus, that would be the spheritorus. Spheric Prism and Toric Prism would be more accurate. Same thing with the glominder in 5D and probably some others.

Thanks
ndl
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### Re: Toratope Names

Hi, ndl. Welcome.

A lot of torotope names are derived from the comb product. There are a lot of swirl toratopes too.
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### Re: Toratope Names

Hi ndl,

The suffix -inder is used to define an extrusion, or product with an interval. I don't think we have a naming convention for the bisecting rotate (lathing). It could be the prefix cyl-, come to think of it. But, we do have one for the non-bisecting rotate (embed/bundle over the circle) , which is -torus .

So, spheritorus means 2-sphere over the circle, and spherinder means prism of the sphere. Torinder can be built two ways: extrusion of the torus, and cylinder over the circle ( a 'cylindritorus' if you will). But, because of these ambiguous dual meanings, we don't really fuss over names much anymore.

Though, I do see the confusion. For the hypercylinders, we have : cylinder , cubinder , tesserinder , penterinder , etc, to define the repeated extrusion of a circle, which also happens to be equal to bisecting rotations of an n-cube. In addition, the prefix is also similar to the name of that respective n-cube. Definitely sends mixed signals, lol.
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ICN5D
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### Re: Toratope Names

Thank you ICN5D for your clarifications. Maybe you could also explain to me the operations that create the various types of torii. I think I understand what your calling non-bisecting rotate, I imagine it as a double bend of the square: into a cylinder and then the top edges bending towards each other inside (or outside really). But in 4D that seems to only create the torinder and the spheritorus.
The concept of spherating described in the wiki doesn't seem like a valid operation. All the other operations add one dimension to the existing toratope, but spherating adds an arbitrary amount and doesn't always represent a basic shape, for example spherating a line gets you a cylinder with two half spheres on the end.
ndl
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### Re: Toratope Names

Spheration is something i invented, to describe what happens when people use sticks and balls etc to make models. The sort of thing that Zome does.

If you draw, in a given dimension, the faintest lines representing a shape, it is hard to hold. But if you apply some arbitary thickness to it, then you can hold it.

A straight line, would give a long cylinder, but the two vertices are typically spherated at a different size, gives more like a dumbell shape. The atomium in Brussles ( http://www.atomium.be is an example of spheration.
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### Re: Toratope Names

There is a lot being found right in this very forum, spread out on various threads of discussion.
I myself found it always inspiring to follow, even so I contributed not so much in the field of round shapes.

P. Wright, M. Četrnáct, and W. Krieger mainly invented the first type of constructive symbols,
while P. Pugeau and Quickfur invented a somewhat different type of constructive symbols.

At some time I tried to line out those constructions to myself and started to aggregate the easiest shapes for examplifying survey,
cf. https://bendwavy.org/klitzing/explain/round.htm. Perhaps this might serve to ease your entry too.

--- rk
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### Re: Toratope Names

Thank you everyone for all the resources. I think I understand what spheration is I just have trouble understanding how it fits together with the other toratopic operations. I also was wondering about tapering, starting with a square isn't there a possibility of tapering of only one of the 2 dimensions to end up with the triangular prism? A square pyramid is more like a double tapering (the same way a torus is a double bending). If explained like this could tapering become commutable with the other operations?
ndl
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### Re: Toratope Names

ndl wrote: I think I understand what your calling non-bisecting rotate, I imagine it as a double bend of the square: into a cylinder and then the top edges bending towards each other inside (or outside really). But in 4D that seems to only create the torinder and the spheritorus.

Not exactly. A bisecting rotate is a rotation (surface of revolution) around an n-1D plane, held at the origin. The bisecting plane behaves like a fixed axle of the rotation motion. This makes/adds rounded faces to the shape, leading to a more 'spherical thing'.

A non-bisecting rotate is when you first translate the shape away from origin (in any direction, usually coordinate), so that the origin lies outside of the interior. Then, revolve around the now outside n-1 plane (stationary axle). This motion makes donuts and adds holes to the shape.

ndl wrote:The concept of spherating described in the wiki doesn't seem like a valid operation. All the other operations add one dimension to the existing toratope, but spherating adds an arbitrary amount and doesn't always represent a basic shape, for example spherating a line gets you a cylinder with two half spheres on the end.

Spheration is tricky to explain. Think of it as more of a symbolic manipulation, that has a few different geometric and algebraic definitions, depending on what's being spherated. I trust that you've seen the open toratope notation symbols, like : (II)I , (II)(II) , (II)(II)(II) , III , etc. To spherate these shapes, simply add another pair of parentheses to the outside : ((II)I) , ((II)(II)) , ((II)(II)(II)) , (III) , etc. These are now the closed toratope forms.

ndl wrote:I also was wondering about tapering, starting with a square isn't there a possibility of tapering of only one of the 2 dimensions to end up with the triangular prism? A square pyramid is more like a double tapering (the same way a torus is a double bending). If explained like this could tapering become commutable with the other operations?

Tapering is meant to be a dimension adding operation, that's like a modified extrusion. So, instead of simply dragging a shape across n+1 dimensions, tapering means to shrink it to a point while dragging. Or, another way is to connect all points of a shape to a single point, into +1 higher dimension (convex hull with a point). But, if you assume unit edge shapes, a tesseract pyramid cannot extend into 5D, because of the whole degenerate pyramid height problem (derived through the pythagoras and long diagonal equations), and so remains in 4D, even though you've technically tapered it.

As far as I know, tapering is only commutable with the bisecting rotate. You can think of the two together as one operation that adds +2 dimensions. A cone is the only 3D example. You could either spin a line into a circle, then taper to cone, or taper the line into a triangle, then spin into a cone. Either way works. The extrude also commutes with bisecting rotate, so shapes like IO>IIO (6D cone x cylinder), have 4 forms : IO>IIO , I>OIIO , IO>IOI , I>OIOI , that are equal. But, extrude does not commute with taper.
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### Re: Toratope Names

ICN5D wrote:Spheration is tricky to explain. Think of it as more of a symbolic manipulation, that has a few different geometric and algebraic definitions, depending on what's being spherated. I trust that you've seen the open toratope notation symbols, like : (II)I , (II)(II) , (II)(II)(II) , III , etc. To spherate these shapes, simply add another pair of parentheses to the outside : ((II)I) , ((II)(II)) , ((II)(II)(II)) , (III) , etc. These are now the closed toratope forms.

Ok, after reading more carefully I think I understand the spheration described in the wiki as the "rss" operation, it's not something that adds dimension at all just rounds out what you already have by extrusion. It's interesting how the cylinder rounds out into a torus.

What I was describing about bending a shape in a new dimension to create a new shape, I think it's a new way way of looking at it not described before but accomplishes the same thing as the bisecting and non-bisecting rotate.

ICN5D wrote:Tapering is meant to be a dimension adding operation, that's like a modified extrusion. So, instead of simply dragging a shape across n+1 dimensions, tapering means to shrink it to a point while dragging. Or, another way is to connect all points of a shape to a single point, into +1 higher dimension (convex hull with a point). But, if you assume unit edge shapes, a tesseract pyramid cannot extend into 5D, because of the whole degenerate pyramid height problem (derived through the pythagoras and long diagonal equations), and so remains in 4D, even though you've technically tapered it.

As far as I know, tapering is only commutable with the bisecting rotate. You can think of the two together as one operation that adds +2 dimensions. A cone is the only 3D example. You could either spin a line into a circle, then taper to cone, or taper the line into a triangle, then spin into a cone. Either way works. The extrude also commutes with bisecting rotate, so shapes like IO>IIO (6D cone x cylinder), have 4 forms : IO>IIO , I>OIIO , IO>IOI , I>OIOI , that are equal. But, extrude does not commute with taper.

Yes, the way tapering is described here would not commute with extrude, but I was trying to define it differently. Instead of just the idea of tapering to a point, what about tapering to another polytope of less dimension than the starting one. The simplest is a square to a line which makes a triangular prism. Also a cube to a line (square pyramid prism) or cube to square (3,4 duoprism). These should all be valid operations. So the old tapering of a square is really a double tapering losing two dimensions at once. If these tapering operations are split up like this then it should commute with everything. As well as creating torii with double bending as I described earlier.
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### Re: Toratope Names

rss and spheration are two entirely different things.

The spheration term was devised to assist modelers describe the polytope model made. Although it involves making round things, it is different to rss.

rss is 'root-sum-square', it is applied to the radial coordinates of several perpendicular bases. It gives a sphere from three lines, each from +1 to -1 on different axies. The centre is taken as 0, and the surface as 1. A radial solid is supposed to consist of concentric shells, where a ray through a given point of that shell is a constant r. Then the particular product of a figure gives R=f(r_x,r_y,r_z).

So if we suppose the bases are lines from -1 to +1, on each of the three axies, then 'R = max(x,y,z)' will give a cube, 'R=sum(x,y,z)' gives an octahedron, and R=rss(x,y,z) gives a sphere. The notation along the line is [max], <sum> and (rss), represent the diameters of a cube, octahedron, and sphere respectively. One can then construct various figures using these.

[iii] = cube.
[i(ii)] = cylinder
(i[ii]) = crind (intersection of two cylinders of the same diameter)
(iii) = sphere. (for different lengths, an ellipsoid)

In four dimensions, you end up with quite a few more.

[(ii)(ii)] duocylinder = bi-circular prism.
<(ii)(ii)> bi-circular tegum
[(iii)i] spherical prism.

One might note that the crind (i[ii]) has 90° edges. It is basically the same shape as a globe, but as if each line of lattitude is replaced by the square surrounding a circle. At four points, you are going to have some pretty sharp corners. Spheration, in one of its forms, softens all sharp angles.

Spheration

Spheration is a kind of surface paint added to models to enhance their appearance, or allow further mathematics. The spheration of three crossing lines, for which the rss (crind product) is a sphere, looks like a jack at https://www.wikihow.com/Play-Jacks . The essential three crossing lines are visible, as are the vertices, all of these being thickened to make them useful to hold.

Other kinds of surface-paint are things like 'surtope paint', where curved surfaces are replaced by small flat planes that follow the surface. This gives a solid to which one can apply polytope rules. A cylinder might become for example, a polygonal prism.
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### Re: Toratope Names

wendy wrote:rss and spheration are two entirely different things.

So when it says in the wiki on the page of toratopes:
Hi.gher. Space Wiki wrote:Each "I" represents a variable and parentheses represent the root-sum-square operation; the toratope is then the surface defined by the resulting polynomial. For example, (II) is the circle √(x2 + y2) − r = 0.

and on the page about toratopic notation:
Hi.gher. Space Wiki wrote:Each vertical line (written as the capital letter I) represents a digon and parentheses represent spheration.

That is a total mistake?
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### Re: Toratope Names

The quote is indeed correct: rss is -root-sum-square-, ie r² = x²+y²+z² for a sphere. This is the rss product of three perpendicular axies (III).

It is not called spheration, but the crind product, as in (rss = crind), <sum = tegum> and [max = prism].

Sheration can be thought of "line-fattening", as in the jacks.

The wiki should have called it the 'crind' rather than 'spheration'.
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### Re: Toratope Names

ndl wrote:Ok, after reading more carefully I think I understand the spheration described in the wiki as the "rss" operation, it's not something that adds dimension at all just rounds out what you already have by extrusion. It's interesting how the cylinder rounds out into a torus.

RSS is a small part of some of the changes you have to do to an equation, in order to spherate it. Again, it depends on what you're spherating. Some examples:

1. (II)I
- the cylinder , disk * line
- spherating makes ((II)I) , the torus , circle over circle

2 Different Geometric changes:
- bend a cylinder into a circle, and glue the circle faces together
- roll cylinder back around itself, aka 'sock rolling ' , and glue circle faces

Algebraic Differences:
- cylinder (II)I :
|√(x²+y²) - z| +|√(x²+y²) + z| - a

- torus ((II)I) :
(√(x²+y²) -a)² + z² - b²

2. (II)II
- the cubinder , disk * square
- spherating makes ((II)II) , the spheritorus , sphere over circle

Geometric changes:
- hollow out the 2d disk into a 1d circle, round and fold 2d square into 3d sphere

Algebraic Differences:
- cubinder (II)II :
|2√(x²+y²) - |z-w|-|z+w|| +|2√(x²+y²) + |z-w|+|z+w|| - a

- spheritorus ((II)II) :
(√(x²+y²) -a)² + z² + w² - b²

3. (II)(II)
- the duocylinder , disk * disk
- spherating makes ((II)(II)) , the tiger , circle over product of two ortho circles

Geometric changes:
- hollow out the 3d faces, so all that's left is the 2d flat torus edge, embed a small circle into every point to 'thicken up' into tiger

Algebraic Differences:
- duocylinder (II)(II) :
|√(x²+y²) - √(z²+w²)| + |√(x²+y²) + √(z²+w²)| - a

- tiger ((II)(II)) :
(√(x²+y²) -a)² + (√(z²+w²) -b)² - c²

4. (II)(II)(II)
- the triocylinder , disk * disk * disk
- spherating makes ((II)(II)(II)) , the triger , sphere over product of 3 ortho circles

Geometric changes:
- hollow out the 5d faces, so all that's left is the 3d flat 3-torus edge, embed a small sphere into every point to 'thicken up' into triger

Algebraic Differences:
- triocylinder (II)(II)(II) :
||√(x²+y²)-√(z²+w²)|+|√(x²+y²)+√(z²+w²)| -2√(v²+u²)| + ||√(x²+y²)-√(z²+w²)|+|√(x²+y²)+√(z²+w²)| +2√(v²+u²)| - a

- triger ((II)(II)(II)) :
(√(x²+y²) -a)² + (√(z²+w²) -b)² + (√(v²+u²) -c)² - d²

5. III
- the 3-cube , line * line * line
- spherating makes (III) , the 2-sphere

Geometric changes:
- evenly round the flat faces and pointy corners of cube into smooth sphere

Algebraic Differences:
- cube III :
||x-y|+|x+y| - 2z| + ||x-y|+|x+y| + 2z| -a

- sphere (III) :
x² + y² + z² - a²

So, as you can see, it really depends on the shape. Spherating is an easy manipulation for some shapes like III -> (III) , but much more complex for (II)II -> ((II)II) .

ndl wrote: Instead of just the idea of tapering to a point, what about tapering to another polytope of less dimension than the starting one. The simplest is a square to a line which makes a triangular prism. Also a cube to a line (square pyramid prism) or cube to square (3,4 duoprism). These should all be valid operations.

They are indeed valid! That's called the convex hull, when you join or 'lace together' two surfaces across a higher dimension. It's a great way to visualize simple shapes like pyramid prism, since they can have multiple convex hulls, which can be seen from rotating the 3D projection of one. Pyramid prism, in particular, can be made by joining a pyramid to a pyramid ; cube to line ; or triangle prism to a square.

R. Klitzing adopted the symbol || (the double pipe) to represent a convex hull of surfaces. So, pyramid prism = pyramid || pyramid ; cube || line ; triangle prism || square . Then you've got more complex ways to join surfaces, like a tetrahedron built from two perpendicular line segments laced across a 3rd dimension: line || ortho line .
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### Re: Toratope Names

ICN5D wrote:R. Klitzing adopted the symbol || (the double pipe) ...

In fact it just mimics the parallel sign. Those shapes are to be placed within parallel hyperplanes atop each other.
It took its first occurance (in that sense) within my paper on convex segmentochora in 2000.

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### Re: Toratope Names

Ok, I'm starting to make sense of all this. Thanks for all your help.
I did see the paper and discussion about segmetachora, that's a much larger set of shapes that include all sorts of combinations. I was trying to stay within a more limited set like the tapertopes. However, after further examination of my version of tapering I discovered that it still is not fully commutable with extrusion so not so useful (but it would add another shape to the set, namely the digonal gyrobicupolic ring as a partial tapering of either square pyramid or triangular prism).
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### Re: Toratope Names

Really it seems the 4||tet (digonal gyrobicupolic ring) belongs with the tapertopes, it's the only other polychoron with only 3D tapertope cells that is not already in the list of tapertopes. Maybe there's another clever way of creating it with existing operations?

Edit: Actually not true there's the 16 and 600 cell and probably others (K.13). But it still seems that this one belongs in the group.
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### Re: Toratope Names

Klitzing wrote:There is a lot being found right in this very forum, spread out on various threads of discussion.
I myself found it always inspiring to follow, even so I contributed not so much in the field of round shapes.

P. Wright, M. Četrnáct, and W. Krieger mainly invented the first type of constructive symbols,
while P. Pugeau and Quickfur invented a somewhat different type of constructive symbols.

At some time I tried to line out those constructions to myself and started to aggregate the easiest shapes for examplifying survey,
cf. https://bendwavy.org/klitzing/explain/round.htm. Perhaps this might serve to ease your entry too.

--- rk

Just as a note (as you also have it on that linked page), it's "Čtrnáct", no "e" there

There's also a "douprism" instead of "duoprism" on the page.
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### Re: Toratope Names

@Marek14:
both is now corrected - at least within my offline copy (which hopefully gets uploaded soon …)
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### Re: Toratope Names

The original notation for spheres and ellipsoids, using xOo format, is that 'x' represents an increase, and 'o' represents the same as the previous, so

xOo = circle xOx = ellipse
xOoOo = sphere, xOxOo = oblate ellipsoid, xOoOx = prolate ellipsoid xOxOx = general ellipsoid.
&c.

The 'dual' is formed using m, where an m is a decrease (initially from infinity). So mOmOo is the radial inversion of xOxOo,

The other thing to note is there may be some confusion between Keiji's brick polytopes (which are by general products of lines), and In5cd's torus-polytopes.

The brick polytopes are simply products of the line segment [-1,1], the result giving the canonical versions of the cylinder, cube, sphere etc. They are necessarily convex, and run along the line, where | or 1 is the absolute value of the line segment (ie abs(-x)=abs(+x)=x), and the surface is defined at x=1, then the shapes are found by (rss), <sum> and [max]. So a cylinder is [1(11)], while the crind or intersection of cylinders or Steinmetz-solid is (1[11]).

In a brick product, (1(11)) = (111) = sphere.

I know that Phil's notation runs to nested circles in a torus, as the tiger be ((11)(11)), being two equal circles in product, spherated by an outer circle. [Spheration and the actual comb product give identical results here, but the comb product applied here does not give an intervening torus, i think. The torotope notation has room for 1's outside the brackets, Judging by Richard's page (11)1 = cylinder, and (11)>. is a cone.

Of course, Keiji's brick notation can not describe torus shapes, and some of the torus-shapes in Phil's notation have no description in terms of products, even though they are topological equivalent to the comb-products. The tiger, for example, i don't think has a comb product description, which is why i use spheration to make it.

duo- correpsonds to english two and german zwo, if that helps getting the letters in the right order. It's what I use.
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### Re: Toratope Names

Is there a similar notation for hyperboloids? A hyperbola, on the large scale, looks like "X", so we could say

xXo = right hyperbola, xXx = general hyperbola
xXoOo = double-sheet hyperboloid (x-axis), xXoXo = single-sheet hyperboloid (y-axis), xOoXo = single-sheet hyperboloid (z-axis)

An ellipsoid is defined by the equation (x/a)2+(y/b)2+(z/c)2=1. An "o" means that b=a or c=b, and an "x" means b>a or c>b. An "O" means that consecutive terms in the sum have the same sign, and an "X" means they have opposite signs. So "xXoOo" means x2 - y2 - z2=1.

wendy wrote: The other thing to note is there may be some confusion between Keiji's brick polytopes (which are by general products of lines), and In5cd's torus-polytopes.

I think his name is a contraction of "I see in 5D."
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### Re: Toratope Names

Hyperbolae are negative spheres, and are simply noted as such. In essence, one writes xi to make x into -x, and we get

xiOx -x²+y²=1
xiOxOo -x²+y²+z² = 1, ie a one-sheet hyperbola.
xiOoOx -x²-y²+z² = 1 ie a two-sheet hyperbola.

It is interesting that you use division-constants in your formulae. Do I suppose this is a curvature constant. In that light, you might note, eg

x²/a + y² / b + z² /c = 1, is a general sphere, but where a, b, or c is negative, turns into a hyperbolae.
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### Re: Toratope Names

Has anyone ever described the shape circle||torus? It's like a cone version of a torinder. What would be a name for it?
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### Re: Toratope Names

It's a cone-circle comb. No one has really looked it it.
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### Re: Toratope Names

ndl wrote:Has anyone ever described the shape circle||torus? It's like a cone version of a torinder. What would be a name for it?

Yes, circle || torus is the cone torus, aka cone-bundle over the circle, or a non-bisecting rotation of a cone.

Equation is:

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

or

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

a > b

a = major radius of the cone torus
b = minor radius of circle base of the cone
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### Re: Toratope Names

If I got your symbols right, then that cone torus, aka circle || torus, ought be
|o>(o) = |>o(o)
ain't it?
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### Re: Toratope Names

Klitzing wrote:If I got your symbols right, then that cone torus, aka circle || torus, ought be
|o>(o) = |>o(o)
ain't it?
--- rk

Yes, you're right!
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