Pentaspace figures

Higher-dimensional geometry (previously "Polyshapes").

Pentaspace figures

Postby ❀りん595❀ » Mon Mar 10, 2008 6:31 am

PLEASE REPLY!!!


pentaspace (n) - five dimensional space
pentaglome (n) - A 4-sphere; the set of points in pentaspace a specified distance from a point
tesserinder (n) - A 5-D figure which is made from a cubinder extended into the fifth dimension
duocylindrical prism (n) - A 5-D figure which is made from a duocylinder extended into the fifth dimension


Suppose Pentanne, a 5-dimensional being, can see Emily's tetraspace living room from her own pentaspace living room. Pentanne can place a tetracube on Emily's living room table and rotate the tetracube through pentaspace to move the tetracube into a position that is impossible with tetraspace alone. Amazing, huh?
I could make this even more amazing. Imagine one day that Pentanne went out on a road on the pentaglome-shaped planet she inhabits. The surface of the pentaspace roads is 4-D, but those roads go in one long dimension like our world's 2-D surfaced roads do. Pentanne pulls out a tesserinder, the resulting figure from extending a cubinder into a fifth spatial dimension. A tesserinder can only cover a line (read: ray) by rolling, thus Pentanne can easily roll a tesserinder down the road. She also has a duocylindrical prism, the result of a duocylinder extended into a fifth spatial dimension. A duocylindrical prism, like a duocylinder, has two mutually perpendicular round sides, but has a *flat* side, like a cubinder. A duocylindrical prism can only cover a line while rolling, and with only rolling on a path parallel to the road, the duocylindrical prism won't go off the road without passing the terminus of the road, as long as the road stays straight.
Now, for the really strange part...
After Pentanne took her rotatera down the roads, she went back home and decided to place her cubinder through Emily's tetraspace. Pentanne took out her tesserinder and put the flat side on Emily's room. Emily then saw the cubinder cross section, and thought, "Whoa! This seems like something I did to Bob with my glome!" Pentanne keeps pushing in the tesserinder further, and while she does this, Emily sees no change in the cubinder cross section. However, a moment later, the cubinder disappears! What actually happened was Pentanne taking the cubinder out of Emily's tetraspace. Pentanne then put in the tesserinder with the *round* side. Emily noticed something different. She saw a cubic hyperprism emerge from zero trength to a full-trength tetracube as Pentanne put in the tesserinder. Then, Pentanne slowly took out the tesserinder, and Emily saw the trength of the hyperprism shrink, and then after that the hyperprism cross section disappeared.

Two notes I would like to see in discussion of this story:
1. How accurate do you think this information is?
2. What do you suppose would happen if Pentanne put in the duocylindrical prism in Emily's tetraspace?
Please share your thoughts and opinions in your replies to this post.

Edit by Hayate: I changed your "tesseractinder" to a "tesserinder" (the proper name) and your "duocubinder" to a "duocylindrical prism" (to avoid confusion, since a duocubinder would usually refer to a 6-dimensional shape). Please don't take offense.
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Re: Pentaspace figures

Postby Keiji » Thu Mar 13, 2008 11:11 am

Well, yes... in general that is simply extrapolating the basic 4D stuff we know into 5D. I will answer your questions though. Firstly, your information is indeed correct, unless I missed something. Secondly, if you take cross-sections of a duocylindrical prism on Cartesian axes, you would either get a duocylinder or a cubinder (since the duocylinder 221 minus the line 1 gives either the duocylinder 22 or the cubinder 21).

By the way, nice username, Rin :P
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Re: Pentaspace figures

Postby ❀りん595❀ » Thu Mar 13, 2008 10:25 pm

Hayate wrote:I changed your "tesseractinder" to a "tesserinder" (the proper name) and your "duocubinder" to a "duocylindrical prism" (to avoid confusion, since a duocubinder would usually refer to a 6-dimensional shape). Please don't take offense.


Oh okay. I used "duocubinder" because a duocylindrical prism can be made from a duocylinder or a cubinder, and "tesseractinder" because a 4D hypercube is known as a tesseract and a cubic prism is known as a cubinder, so I put "tesseract" and "-inder" together, and I have no idea what a 6D duocubinder would look like. I have some more terms, so correct the terms around parentheses as needed.

Done: spherindrical prism -> spherical diprism, and duospherinder -> cylspherinder. Right now I wish I still had my geometry wiki... Also, try out the quote tags sometime ~Hayate


spherical diprism >> A hyperprism with a spherindrical base {3+1+1}; also possible by rotating a cubinder into pentaspace
cylspherinder >> Cross product of a sphere and a circle
glominder >> Glome prism
pentaglome >> 4-sphere; 5D hypersphere
pentacross >> 5D cross polytope

In addition to a tesserinder and a duocylindrical prism, Pentanne has five other 5D shapes. A spherical diprism can cover a plane with only rolling, and has two linear axes, like a cubinder. Pentanne puts the spherical diprism into Emily's tetraspace, and then Emily sees a spherindrical intersection with unchanging size. Then, when Pentanne takes out the spherical diprism, Emily sees a spherinder disappear! That's from a flat side of the spherical diprism.
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Re: Pentaspace figures

Postby wendy » Tue May 27, 2008 8:29 am

The PG name for five dimensions is horopetix, derived thus:

horo: horizon-centred, = euclidean + peton, petix: 5d patch, cloth.

Polytopes formed in this space hight "poly-ter-on" = "many (closed) + 4d + patch"

I am not sure what the assorted words Hayate etc use here, but there are three active products in five dimesions. The adjectival use suggests that root elements (line, line, point), make the missing element, eg

pentagonal prism = pentagon × line prism
pentagonal tegum = pentagon × line tegum
pentagonal pyramid = pentagon × point pyramid.

In five dimensions, one might have, eg circle-circle pyramid, or circle-sphere tegum or prism.

Names like "pentategum", "pentaprism", "pentaglome", are yet to be given recognised status. It could well be done if the suffixes are allowed to fall in the general stem-class /tope/, where the resulting figure corresponds to that dimension. It here here that we find that these correspond to the fifth powers of the general radiant products sum(), max(), rss(), applied to the 1d line.

The difficulty with the simplex is that the 5d simplex is the sixth root power, ie it is the product of point^6, so using a name like penta- could variously be taken as the fifth power or fifth dimension.

spherical diprism = glomohedron-square prism
cylspherinder = "glomohedron-circle prism" but RHS = "~ tegum"
glominder = glomochoral prism = glomochoron-line-prism.
pentacross = pentategum.
pentaglome = 1. glomoterix (surface) = globe-shaped 4fabric or 2. glomoteron (solid) (globe-thing covered by 4d patch.
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Re: Pentaspace figures

Postby Keiji » Thu Jun 05, 2008 9:43 pm

Hooray, nothing like more of Wendy's confusing terms :mrgreen:

May I ask what you mean by a circle-circle pyramid? Pyramid isn't a product, we had a topic debating that ages ago, and IIRC we came to that conclusion. The three products are always max/sum/rss or prism/tegum/crind for any dimension, though they behave somewhat oddly in dimensions less than three.
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Re: Pentaspace figures

Postby wendy » Fri Jun 06, 2008 7:24 am

Pyramid is indeed a product of general products.

One needs to look at the surtope consist of the simplexes in N dimensions to see the pascal triangle.

The "circle-circle pyramid" exists in five dimensions. For circles of equal size, we have

s = (v-v1)rss(w,x)+(v2-v)rss(y,z) Surface is when s=1

The product as described is an intersection of two five-dimensional wedges, each with 2d tips (rss(w,x), and rss(y,z)).

v is the altitude of height of the pyramid (which can be any dimension). The apices are at v1, v2. In four dimensions, perpendicular to the v-axis, we have a prism-product of circles, a.rss(w,x) * (1-a)rss(y,z). This looks like a duocylinder, of two circles of bases that add to a constant measure.

Another way of visualising it, is to consider first the triangle (with height in v) This is point ** line

If you increase the dimension to three, you can replace 'line' by circle, to get a cone (point ** circle), or you can replace the point by a line vertical to the plane (ie line ** line).

Increasing the dimension again gives (line ** circle), eg a 'duo-cone', or conic pyramid

Increasing the dimension again gives (circle ** circle), ie bi-circular pyramid.

Like all pyramids, it multiplies both the surtope consist with 1's at either end, (ie include both content and nulloid).

circle = h+e+n (area, line, no points, nulloid) = 1,1,0,1

bi-circular cone = (1,1,0,1)² = 1,2,1,2,2,0,1 = 1p+2t+1c+2h+2e+0v+1n

This gives
1x = peton (6d = content)
2p = 2 terons = line*circle pyramid (line = circle-surface)
1t = 1 choron = tetrahedron (opposite edges bent into circles)
2d = 2 circles = bases
2e = 2 circle cumfrences
0v = 0 vertices
1n = 1 nulloid.
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Re: Pentaspace figures

Postby ICN5D » Mon Jul 28, 2008 4:30 am

If you are the creator of the HDD(higher dimensions database), then I would like to know more about your systematic way of finding the names and notations of all the strange, exotic shapes . I have noticed that the rotope construction chart goes up to five dimensions, and there is a systematic way of building them. Then I noticed there was strange shape called the cylcyltigerinderindric torus and had number 30117. This is why I really would appreciate it if I could learn more. It seems as if there may be a computer program that is computing all of these crazy shapes. Because, I mean come on, who's gonna sit there and calculate thirty thousand different objects? I also have a strong interest in the linguistic side of it, too. Any help with that would be icing on the cake. And again, my email is philociraptor@gmail.com
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Re: Pentaspace figures

Postby wendy » Tue Jul 29, 2008 8:32 am

I do not maintain the HDD, but i do maintain the "polygloss". Much of the PG is devoted into a uniform naming system, entirely different to the existing system.

The current coverage of the PG is the uniform polytopes, their duals, and kindred processes. Some other solids are mentioned.

The current naming convention for uniform polyhedra is to tokenise the underlying stott-wythoff vector form into letters and numbers, using prism product for abutal, eg "ACd1" is a product of A (a line), and Cd1 = 3d 53 xoo, ie x5o3o = dodecahedron => gives dodecahedral prism in 4d.

There are some notations for the circles, by reading the circle, sphere &c as polytopes O, OO, OOO &x. Applying middle truncations gives rise to the ellipses.

For the pyramid product, no formal notation exists, except to treat them as a lace-product over several bases, eg xoOoo&oxOoo&x gives the bicurcircular pyramid in 5D. Likewise, we have the \O&\O = bicircular tegum, dual to /O&/O bicircular prism.

A term like Dd2 does allow one to find the assorted metrical properties, since 'd' gives rise to a vector (f, 2, 3-f. 4-f, ...), and an 'animal' n, which allows the creation of the stott matrix. '2' allows the stott vector 0,1,0,0 to be created [ie 1, 2, 4, 8], and the matrix dot with the stott matrix over two copies of the stott vector allows one to determine the diameter of the figure.

Many of the names by Bowers, Johnson, &al are simply implememtations of the stott vector anyway.

There is a system here that uses <> for the tegum product, [] for the prism product, and () for the circle-product, so that one might write something akin to <(wx)(yz)> for the bicircular tegum. The dual is [(wx)(yz)].

I am not sure what 30117 might mean, but it is not necessary that there is 30116 previous examples: it might for example, be a stylised construction, like ACd1.

Pentaspace (PG = petix) in particular gives no special meaning, the distinction is made between 5D (solids), and 5T (tilings), because the latter function as 6D solids. There is supposed to be something pretty cute about the group 5g, 5y, but i have not uncovered it.
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Re: Pentaspace figures

Postby ICN5D » Wed Jul 30, 2008 2:53 am

The number 30117 stood for the order in which that shape was in, in the HDD. The whole Rotope Construction Chart is pretty much a graphical tree that shows the order of the permutations. Anyone could eventually draw the chart out, but the program that computed all those shapes is what I'm after. The nomenclature used in the chart also appeals to me for its systematic nature. The notation, once understood, allows you to decode each transformation and greatly aid in visualizing it.
http://teamikaria.com/wiki/Rotope_construction_chart

I've recently grasped the concept of the circle product. I had the square product down, simple, just a twice-extruded prism. But the circle was more elusive. Then it dawned on me one morning, while I was sitting in my car, that a circle creates an extruded prism that has varying height! A duocylinder is the cartesian product of two circles. It's 3-D cross-sections are cylinders of varying heights. That makes sense, because the cross-sections of a circle are lines of varying length. A circle product adds an extruded prism to a shape and where the circular part comes into play is the varying height of the cross-section prisms. So, a cyltrianglinder, which is the CP of a triangle with a circle, makes a triangular prism of varying heights! When you take a slice at the top of the circle, the prism is thin. Moving down makes the prism grow, then stop, at the middle. Moving further will flatten, then dissapear. So, if I extrapolate this further, then a sphere product will create a duoprism of varying lenghth+width, or height+length, etc. This should not be mistaken for the taper product, which creates a fixed prism of various sizes.
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Re: Pentaspace figures

Postby ICN5D » Fri Aug 01, 2008 1:15 am

today was the first day that I was able to comprehend a ten-dimensional object. It's notation goes like this :

((II)(II)I)I'(II)I or ((xy)(zw)v)u^t(sr)q

This complicated-looking shape can be dissected for better understanding. Since there is a torus product ((n)I) , we'll start with that one : inside the torus, we find a duocylinder (II)(II) , making a duocylinder torus ((II)(II)I) . This torus is now mult by a triangle I' . A simpler way to express a triangle product nI' is to convert it into an extrusion I that tapers down to a point ' (this is essentially what a triangle is). The action on the torus is now to extrude into a cylinder (n)I , and taper to a point making a cone (n)' . This torus + cylinder + cone can be called a torindricone ((II)I)I' . The last product is a plain old cylinder (II)I . This means that a duocylinder torindricone is mult by a cylinder. Another way of saying this is that the surface of the cylinder is a 7-D manifold of a duocylinder torindricone. So the 10-D shape is a duocylinder torindricone manifold of a cylinder. See what I mean, not too complicated!
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Re: Pentaspace figures

Postby Keiji » Sun Aug 03, 2008 10:17 pm

Oh wow, I should stop leaving for ages. Sorry about that! Anyway, I am the creator of the HDDB (HDD stands for hard disk drive ;) ), so I shall now answer your questions:

I would like to know more about your systematic way of finding the names and notations of all the strange, exotic shapes.


Names are decided on a heuristic basis for the most part. Shapes that already have common names were just kept the same, and the higher-dimensional ones were formed from them. This mainly consists of adding the words "prism", "pyramid" or "torus" and changing previous words to adjectives ("prism" -> "prismidal", "pyramid" -> "pyramidal", "torus" -> "toric", shape adjectives usually ending in "al", "ar" or "ic"). Where a word is repeated, number prefixes are used, so a "prismidal prism" is a "diprism", a "dipyramidal pyramid" is a "tripyramid" and a "tritoric torus" is a "tetratorus" (note that "di" != "bi", a "dicone" is a 4-dimensional object formed by tapering a cone, whereas a "bicone" is a 3-dimensional object formed by taking the tegum of a circle). For the strange rotopes, the naming system can break down. The Cyltrianglintigroidal pyramid is an example of where I simply formed the best fitting name I could, and I believe there was another rotope I found that I could not name at all.

As for notations, the rotopic notation consists of characters (traditionally dimension letters but bars are used on the construction chart as they look cleaner and distinguishing the dimensions is unnecessary) grouped by zero or more pairs of parentheses and with zero or more superscripts. Parentheses indicate spheration whereas superscripts indicate tapering. Juxtapositions indicate mere extrusion into another dimension or Cartesian product. The notation of rotopes has quite a history if you're interested - to summarize it briefly it began with Alkaline's "rotatopes" which used only non-nested parentheses and juxtapositions; subsequently "toratopes" were added which had nested parentheses and gave rise to the strange rotopes; most recently the "tapertopes" which added superscripts and similarly gave rise to the ambiguous and immeasurable rotopes, and also introduced orientation distinction in the notation which I dislike in retrospect, but there is little better way to do it.

Then I noticed there was strange shape called the cylcyltigerinderindric torus and had number 30117. This is why I really would appreciate it if I could learn more. It seems as if there may be a computer program that is computing all of these crazy shapes.


I had written a PHP script to generate all rotopes up to a given number of dimensions, which was where I got the number 30117 from, and called it the last 9D rotope. However, I am aware the number is incorrect (though the notation for the rotope on the wiki page is indeed describing the last 9D rotope) as I failed to notice there is a problem with the script in dimensions above five, which I simply never got around to fixing, and also never got around to mentioning that on the wiki.
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Re: Pentaspace figures

Postby ICN5D » Wed Aug 06, 2008 1:18 am

Cool! I'm glad you responded. It seemed that the HDD was a dead website. Too bad because I have a hunger to learn more about this. I could never have imagined that I could comprehend such things. I'm sure that you've seen my other posts. All I new well were the n-cubes until I came across your notation. The language of it paints pictures in my head. And please tell me if my grasp of the 10-d object is correct, I'd like to know if I'm really gettin' it. I also wanted to ask you if the names are computer generated, too. And, I especially like the cleverness of the torus notation ((II)I) it makes complete and total sense to me : a circle (II) extruded along the path of a circle(nI). I've also take it to myself to add another function to the list : the spin, and I found some intriguing mathematical properties in the sequence of shape operations. By adding the spin to the construction sequence, the individual operations themselves had a commutative property for most of them. This meant that for certain combinations of operations, it didn't matter which order they were in, it made the same object! This special property allowed for a deeper insight into the workings of higher dimensions. For instance, consider a cylinder (II)I . It can be made two different ways : extruding a circle or spinning a square. If we change (II) to I+ , where + stands for the spin function of a line, by rewriting (II)I to I+I , it's now easier to see the square and the circle. So I+I can also be II+ and they both make the same shape! Cool, huh? Take the duocylinder now : a circle times a circle (II)(II) ==> I+I+ . The +I is ambiguous and can be reordered I+I+ ==> II++ and alas, we find that the spin of a cylinder also creates a duocylinder. But note however that I+I+ cannot be reordered I++I , because this is a spherinder. If you're more interested in this, please let me know, because I think it's pretty clever.
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Re: Pentaspace figures

Postby Keiji » Wed Aug 06, 2008 11:40 am

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


This complicated-looking shape can be dissected for better understanding. Since there is a torus product ((n)I) , we'll start with that one : inside the torus, we find a duocylinder (II)(II) , making a duocylinder torus ((II)(II)I) . This torus is now mult by a triangle I' .


This part is wrong. A superscript applies to everything before it - the six-dimensional rotope ((II)(II)I)I is tapered to a point to form the seven-dimensional rotope ((II)(II)I)I'. This is different to the Cartesian product of a triangle I' and the five-dimensional rotope ((II)(II)I), which is I'((II)(II)I). As I mentioned in my last post, the superscripting creates differences in positioning sub-parts of the notation, which I myself am not particularly happy with.

triangle product


There is no such thing as a triangle product! This is why your understanding is incorrect: there is a Cartesian product formed by juxtaposition, a spheration operation formed by surrounding with parentheses and a taper operation formed by appending a superscript. An operation is not the same as a product! An operation takes a specified number of arguments (in the case of the taper and spheration operations, one argument) and when there are multiple arguments the result differs depending on the order. A product takes any nonnegative number of arguments and gives the same result no matter what the order is.

The rest of your interpretation is correct, though. Your duocylinder torindricone would be systematically called a duocylindrical toric prismidal pyramid.

I also wanted to ask you if the names are computer generated, too.


The PHP script I mentioned does also name the rotopes, but I usually just name them manually under the systematic conventions anyway.

And, I especially like the cleverness of the torus notation ((II)I) it makes complete and total sense to me : a circle (II) extruded along the path of a circle(nI).


Then you are saying (II) means to extrude a line along the path of a line? ;)

The spheration operation never made much sense to me, I must admit - it was designed by someone else on this forum. I only have a clear picture of toratopes of the form ((x)I), where x is another rotope.

spin


Frankly, this is a mess. It's already been done in the archaic notation for rotatopes, where every character was a number denoting an n-sphere (1 = line, 2 = circle, 3 = sphere, etc) and the rotatope was the Cartesian product of every character. There, your I was a 1, I+ or +I was a 2, I++/+I+/++I was a 3, etc. This easily explains the ambiguity you had, as I++I could be written as 31 or 22, where 31 is a spherinder and 22 a duocylinder. The entire point of the rotopic notation was to make it unambiguous.

As I've said though, tapertopes do screw it up and make it rather unintuitive. I'm thinking it would be a better idea to use another surround for tapering, rather than superscripts. Thus:

<II> = triangle
<III> = square pyramid
<IIII> = cubic pyramid
<II><II> = Cartesian product of two triangles (not representable in current notation)
<(II)I> = cone
<<II>I> = tetrahedron
etc.

Of course this now asks the same questions as spheration initially did: what is <<II><II>>, for example? ((II)(II)) is called a tiger, maybe we should call <<II><II>> a lion? :mrgreen:

Using a surround does solve the orientation problems though. Also, using this would increase the number of rotopes in dimensions above three, as you can see from the examples above, so I would have to reorder them all again - I think I'd like having <II><II> as a rotope, though, as it's a nice shape.

You might also be interested in the bracketopes, which use three products and no operators, making it much more intuitive.
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Re: Pentaspace figures

Postby ICN5D » Sun Aug 17, 2008 3:05 am

Then you are saying (II) means to extrude a line along the path of a line?
No, (II) means to extrude a line along the path of a circle, which can be represented as (nI) .To see this interpretation in action, consider the torus, ((II)I) . The format of the operations are in (nI), which means that it has the shape of a circle. The tricky part is to apply the concept of a circle for "n" in the generalized circle notation. This makes the torus, ((II)I) , which means to extrude a circle along the path of a cirlce: "n" is the embeded surface object of the circle (nI) . The number of extrusions that follow the surface object denote the dimensionality of the path to follow. So ((II)II) is the toracubinder, which is a circle (II) that has been extruded along the path of a sphere, denoted as (nII) .
There is no such thing as a triangle product!
Yeah, you'll have to excuse my earlier nomenclature on this subject. My personal interpretation of an extrusion followed by a taper is to condense the two operations into a more elegant form of saying a triangle operation. Still keeping in mind that the order of operations is not commutative, the same can be applied to the double extrusion being called a square operation.
But something peculiar happens when one considers a circle operation. The first time I attempted to understand the circle operation was when I came across the notation for the cyltrianglinder I'(II) . This is the triangle times a circle. Now I wondered what the heck was this shape and how do I underdstand it. For a better idea I looked to the cylinder (II)I , or I(II) . At this point, I had introduced the spin operation "+" and I'll show you why I like it better : The circle can be defined as a line that has been spun in place along the 2nd dimension : "I+" . So the (II)I would look like I+I . As I mentioned in my earlier posts, I+I can also be II+ . Now just think of the implications of this new notation : it says that a cylinder can be made by extruding a circle I+I OR by spinning a square II+ , which is exactly true. Another neat one is the cone I+> (if you don't mind let's call > the taper). So the cone can be made two different ways : by tapering a circle I+> OR by spinning a triangle I>+ . The > and + are ambiguous! Now, back to the cyltrianglinder. After seeing how a circle operation can be reduced into an extrusion and a spin, I'(II) can be rewritten as I>I+ (notice how the order is unambiguous). This is where I totally freaked out. The cyltrianglinder is a triangular prism I>I that has been spun along the fourth dimension! A property that I observed about spinning an N-prism is that the cross sections are N-prisms of various height. Think about it : if you sliced a cylinder lengthwise, you'd get squares of various widths. This holds true to anything with a spin operation : if you spin a triangle prism, you would get cross sections of triangle prisms of various heights. Another cool trick is to look at the cubinder I+II . This is quite clearly a cylinder that has been extruded. Applying the ambiguous property of the last three operators "+" and "I" x 2, I+II can be II+I and III+ , which is also a cube that has been spun! (hence the name cubinder)
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Re: Pentaspace figures

Postby quickfur » Mon Aug 18, 2008 4:32 pm

The product of a triangle and a cylinder is just a triangular cylindrical prism (or 3-prismic cylinder). It is bounded by 3 cylinders folded into a triangle in 4D, and a triangular torus surface. There's nothing scary about this at all. The prismic cylinders are simply the limit of m,n-duoprisms as one fixes m and let n approach infinity (or vice versa). If you let both m and n approach infinity, you end up with the duocylinder.

You can see some figures I made of these neat objects here: http://eusebeia.dyndns.org/4d/priscyl.html
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Re: Pentaspace figures

Postby wendy » Wed Aug 20, 2008 8:33 am

quickfur wrote:The product of a triangle and a cylinder is just a triangular cylindrical prism (or 3-prismic cylinder)


There is no single product: there are several different products, not just one. A triangle #* cylinder gives the triangle-line-circle prism.

A triangle *# cylinder gives a triangle - cylinder tegum. A triangle ** cylinder gives the cylinder tri-point pyramid (the same as taking the point-pyramid three times on the cylinder). The triangle ##cylinder comb gives in four dimensions, a figure derived by taking the surface of a triangle (ie the line), and replacing these by bars made out of circle-line prism section.

There are the radiant products, to which the tegum (sum) and prism (max) have been dealt with above. Also the rss() product gives the crind product.

Speration corresponds to taking a frame (eg vertex + edge), and replacing each point by a sphere of fixed size. This would, for instance convert an edge into a pipe, or a hedron (eg triangle) into a slab with rounded edges. It is not a product but an effect (like truncation, rectification), applied to a framework.
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Re: Pentaspace figures

Postby quickfur » Wed Aug 20, 2008 4:19 pm

wendy wrote:
quickfur wrote:The product of a triangle and a cylinder is just a triangular cylindrical prism (or 3-prismic cylinder)


There is no single product: there are several different products, not just one. A triangle #* cylinder gives the triangle-line-circle prism.

I was referring to the Cartesian product, actually, which is unique. But you're right, there are other possible products which yield different objects.

[...]Speration corresponds to taking a frame (eg vertex + edge), and replacing each point by a sphere of fixed size. This would, for instance convert an edge into a pipe, or a hedron (eg triangle) into a slab with rounded edges. It is not a product but an effect (like truncation, rectification), applied to a framework.

Sounds interesting. I guess the result would depend on the dimension of the sphere, right? So 2-spherating an edge gives a pipe, whereas 3-spherating an edge gives a spherinder, etc..
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Re: Pentaspace figures

Postby wendy » Thu Aug 21, 2008 7:59 am

Spheration is a name given to an effect that makes everything solid. So if you had a 2d structure in 4d, running a glome around it makes it into a solid.

One can differentiate the spherations, to emphersize elements. For example, one might make the vertices into spheres of size 1, but lines of cylinders of size 1/2, and hedrons into lines of size 3.
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