Polyprism, Prisminder, and Rotaprism

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Polyprism, Prisminder, and Rotaprism

Postby Neues Kinder » Fri Oct 28, 2005 11:40 pm

I came up with three new (?) types of 4D figures.

The first is the Polyprism. You can get a 3D n-gonal prism by taking an n-gon and extending it once. Well you can also get a prism by taking a line and extending it x times around in a circular fashion. You can do the same in the transfer from 3D to 4D. You can make a 4D n-hedral prism by taking an n-hedron and extending it once (including prisms, which make duoprisms). You can also make prisms by taking an n-gon and extending it around x times in a circular fashion, or by taking a line and extending it around in a spherical fashion (which only results in an ordinary prism). Like when you take a line and extend it around 3 times you get a triangular prism, you can take a triangle and extend it around 3 times to get what I call a triangular trinikprism. If you take a square and extend it around 3 times, then you get a rectangular trinikprism (the name can be reversed - it is also a triangle extended around 4 times, which would make it a triangular tetrikprism). If you take a pentagon and extend it around 6 times then you get a pentagonal hexikprism (also the hexagonal pentikprism). Why I add the "k" is because I already came up with the triprism (5D - extend the duoprism), the tetraprism (extend the triprism), pentaprism, etc.

The second one I came up with was the Prisminder, which is a rotated prism (rotated the same way that you rotate a cube to get a cubinder or a cylinder to get a spherinder). You can also get a prisminder by extending the circle n times around in a circular fashion.

The third type is called the Rotaprism, which is simply a prism rotated every other way. Like when you rotate both of the opposite squares of the cube with its diagonal as the axis, you get the diconic rotaprism. When you rotate a triangular prism with one of its triangles' medians as the axis then you get a conic rotaprism. When you rotate the pentagonal prism with the axis going from one vertex to the midpoint of the opposite side you get what I call a conidifrustrumic rotaprism (basically a cone and two frustrumes (a trapezoidal cylinder) stacked on top of each other extended, hence its name). When you rotate the hexagonal prism with the axis on a diagonal cutting it in half, you get a diconifrustrumic rotaprism (cone, frustrume (cylinder if it's a regular hexagon), cone), and when you rotate it on the axis that connects the midpoints of both sides you get the difrustrumic rotaprism.
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Re: Polyprism, Prisminder, and Rotaprism

Postby Marek14 » Sat Oct 29, 2005 7:30 am

Your polyprisms are well known. What you call a triangular triniprism is generally known ar triangular DUOPRISM. Rectangular trinikprism or triangular tetrakiprism is called Triangular-square duoprism. In other words, duoprisms are generally considered much broader category - the ones you call duoprisms are just polygon-square ones.

The easiest way to think of duoprism is to imagine it as cartesian produt of two polygons in 4D, one in xy plane and one in zw plane.

Prisminders are not generally considered here, but the name is quite good. These correspond to circle-polygon duoprisms. This means that they don't get any vertices as such, but they have some circular edges and curved mantles.

I'm afraid that I don't quite understand the rotaprisms. If you take anything in 3D and rotate it around an axis, you get something else in 3D. To get to 4D, you'd have to rotate around a plane. (note thet you can imagine rotating around an axis as rotating around a plane perpendicular to the hyperplane where your figure lies, the same as rotating plane around a point is the same as rotating it around a line pernendicular to it)
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Postby wendy » Sat Oct 29, 2005 7:59 am

The rotoprism is simply the circle - polygon prism.

Many of these are simply just prism or sphere-products of existing figures.

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Postby Marek14 » Sat Oct 29, 2005 12:53 pm

So, is there any difference between rotaprisms and prisminders?
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Postby Neues Kinder » Sat Oct 29, 2005 5:32 pm

Sorry about the confusion. When I did the post I forgot that you rotate 3D objects around a plane to make it 4D. I'll try to explain this a different way.

To get the prisminder, you just rotate around all of the lines that connect the corresponding points of both polygons in the prism to make circles. Another way to see it is n cylinders connected end-to-end, forming a circular chain, to form an n-gonal prisminder.

To get a rotaprism, you take an n-gon, and if n is even you either rotate it around an axis through opposite vertices or opposite midpoints, and if it's odd you just rotate it around an axis through a vertex and the midpoint opposite to it, and then extend the resulting 3D figure.

So yes, there is a difference between the rotaprism and the prisminder.
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Postby wendy » Sun Oct 30, 2005 5:50 am

i am not following the drift with converting lines into circles. Making the lines into circle-cross-sections does not work on edges. The way we convert a square to a cylinder is to draw "width" lines, which become base circles.
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Postby Marek14 » Sun Oct 30, 2005 8:04 am

Neues Kinder wrote:Sorry about the confusion. When I did the post I forgot that you rotate 3D objects around a plane to make it 4D. I'll try to explain this a different way.

To get the prisminder, you just rotate around all of the lines that connect the corresponding points of both polygons in the prism to make circles. Another way to see it is n cylinders connected end-to-end, forming a circular chain, to form an n-gonal prisminder.

To get a rotaprism, you take an n-gon, and if n is even you either rotate it around an axis through opposite vertices or opposite midpoints, and if it's odd you just rotate it around an axis through a vertex and the midpoint opposite to it, and then extend the resulting 3D figure.

So yes, there is a difference between the rotaprism and the prisminder.


I see. Then it seems to me that you are skipping a step here. You should first explore the properties of the 3D rotaprism "intersteps" (let's call them rotagons for now), and then you can think of extending it.

As I see it, there are three distinct kinds of rotagons:

1. From even n-gons, axis through vertices
2. From even n-gons, axis through edge midpoints
3. From odd n-gons.

In case 1, the resulting rotagon has 2 special vertices, (n/2-1) circular bands, and (n/2) faces, which look as parts of conic surfaces (two of them, those at the end, ARE conic surfaces). For n=4, the result is isomorphic to a bicone.

In case 2, there are no special vertices and (n/2) circles. Apart from (n/2-1) conical faces, there are two flat, circular ones.

For the same, even n, the case 1 and 2 are dual figures.

Case 3 is a hybrid of cases 1 and 2. There is 1 vertex, ((n-1)/2) circles, ((n-1)/2) conic surfaces and 1 flat circle. These rotagons are self-dual.
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Postby Batman3 » Sun Oct 30, 2005 9:58 pm

Could you expand on the details of case 3 rotagons? Is the line of rotation through a symmetry line?

Is there any way I can visualise this stuff apart from the perpendicular directions(i.e. xy,xz,yz,xw,yw,zw). That is to say from some kind of angle(or bi-angle or tri-angle?) like 30 degrees or 45 degrees?
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Postby Marek14 » Mon Oct 31, 2005 8:08 am

Batman3 wrote:Could you expand on the details of case 3 rotagons? Is the line of rotation through a symmetry line?

Is there any way I can visualise this stuff apart from the perpendicular directions(i.e. xy,xz,yz,xw,yw,zw). That is to say from some kind of angle(or bi-angle or tri-angle?) like 30 degrees or 45 degrees?

Yes, the line of rotation is one of their symmetry lines.

There probably is a way to visualize it in another orientation, but I didn't find it yet.
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Postby Neues Kinder » Sun Nov 20, 2005 10:57 pm

Marek14 wrote:Your polyprisms are well known. What you call a triangular triniprism is generally known ar triangular DUOPRISM. Rectangular trinikprism or triangular tetrakiprism is called Triangular-square duoprism. In other words, duoprisms are generally considered much broader category - the ones you call duoprisms are just polygon-square ones.

I don't believe that the polyprisms are duoprisms. Duoprisms, in my view, are extended prisms, or prismic prisms. A polyprism, on the other hand, is an n-gonal prism folded up (x-1) times, wrapped around an x-gonal prism folded up (n-1) times. What I mean by that is the same as saying that a triangular prism is a rectangle folded 2 times, wrapped around 2 triangles.

The duoprism is related to the cubinder, (1,1,2), and the polyprism is related to the duocylinder, (2,2). The duoprism is a polygon extended 1D twice, and the polyprism is a polygon extended 2D, or in a polygonal fashion. The prisminder is also related to the duocylinder, except that it's either a circle extended 2D, or a polygon extended around in a circle, which can also be viewed as an n-gonal prism wrapped around a cylinder folded up (n-1) times.
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Postby Marek14 » Sun Nov 20, 2005 11:35 pm

Neues Kinder wrote:
Marek14 wrote:Your polyprisms are well known. What you call a triangular triniprism is generally known ar triangular DUOPRISM. Rectangular trinikprism or triangular tetrakiprism is called Triangular-square duoprism. In other words, duoprisms are generally considered much broader category - the ones you call duoprisms are just polygon-square ones.

I don't believe that the polyprisms are duoprisms. Duoprisms, in my view, are extended prisms, or prismic prisms. A polyprism, on the other hand, is an n-gonal prism folded up (x-1) times, wrapped around an x-gonal prism folded up (n-1) times. What I mean by that is the same as saying that a triangular prism is a rectangle folded 2 times, wrapped around 2 triangles.

The duoprism is related to the cubinder, (1,1,2), and the polyprism is related to the duocylinder, (2,2). The duoprism is a polygon extended 1D twice, and the polyprism is a polygon extended 2D, or in a polygonal fashion. The prisminder is also related to the duocylinder, except that it's either a circle extended 2D, or a polygon extended around in a circle, which can also be viewed as an n-gonal prism wrapped around a cylinder folded up (n-1) times.


This is just a question of nomenclature and convention. What I was trying to say is that in this particular area, the convention already exists, and has existed for years. See, for example, http://members.aol.com/Polycell/section6.html and the way it uses the term "duoprism".

I stay by my claim that "duoprism" is standard term for what you call "polyprism" and what you call "duoprism" is just a special case where one part of the duoprism is a square.

On the other thought, it's possible I misread you and you claim that extending a prism and multiplying a polygon with square are two different operations resulting in two different polytopes. To refute that, you just have to consider that a square is a prism product of two lines. Then, the first shape is (polygon*line)*line, and the second is polygon*(line*line). Prism product is associative, so both are one figure.
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Postby Neues Kinder » Thu Dec 01, 2005 3:45 pm

I've been thinking that over, and I came to a different conclusion similar to yours, only backwards. When we classify polytopes we start with a wide-ranged category, then we split that category into smaller ones, then split those into even smaller ones, and keep doing that until we can't be any more specific. Think of polygons. Polygon is a word meaning all 2D polytopes. We then divide those polygons into convex and concave. We eventually arrive at rectangles and squares. A rectangle is a parallelogram, and so is a square. A rectangle is also an equiangular parallelogram, and so is a square. They are exactly alike except for the fact that a square is also equilateral and the rectangle is almost never equilateral. The square is just a more specific definition of a rectangle.

It's the same basic concept with the polyprism and the duoprism. A duoprism, at least my definition of a duoprism, is a prism with one polygonal part and one rectangular part. Whereas a polyprism is a prism with two polygonal parts. A polyprism and a duoprism are almost exactly alike except that a duoprism has one rectangular part and a polyprism almost never has a square part.

Saying that all polyprisms are also duoprisms is just like saying that all rectangles are squares.

They call them duoprisms because they are prismic prisms, or extended prisms.
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Postby Marek14 » Thu Dec 01, 2005 7:27 pm

Neues Kinder wrote:I've been thinking that over, and I came to a different conclusion similar to yours, only backwards. When we classify polytopes we start with a wide-ranged category, then we split that category into smaller ones, then split those into even smaller ones, and keep doing that until we can't be any more specific. Think of polygons. Polygon is a word meaning all 2D polytopes. We then divide those polygons into convex and concave. We eventually arrive at rectangles and squares. A rectangle is a parallelogram, and so is a square. A rectangle is also an equiangular parallelogram, and so is a square. They are exactly alike except for the fact that a square is also equilateral and the rectangle is almost never equilateral. The square is just a more specific definition of a rectangle.

It's the same basic concept with the polyprism and the duoprism. A duoprism, at least my definition of a duoprism, is a prism with one polygonal part and one rectangular part. Whereas a polyprism is a prism with two polygonal parts. A polyprism and a duoprism are almost exactly alike except that a duoprism has one rectangular part and a polyprism almost never has a square part.

Saying that all polyprisms are also duoprisms is just like saying that all rectangles are squares.

They call them duoprisms because they are prismic prisms, or extended prisms.


What I'm pointing out is that in that case, you should pick a different name. "Duoprism" is taken, and it, in fact, covers extremely large category of shapes, since it is used for any cartesian product of two 2D shapes - for example, duocylinder can be also called "circular duoprism".
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Postby Neues Kinder » Mon Mar 06, 2006 2:02 am

I was seeing if I could find the formulas for the rotatopes just to see if I knew them sell enough, and I thought maybe I could come up with formulas for the surface volume and bulk of duoprisms instead. It took me, like, two days to come up with the formulas for all the different types of duotriangular prisms. I classified those prisms by which type of triangle each of the two triangles are (equilateral, isosceles, scalene), and based on how each of their sides are congruent to each other. I eventually came up with these:

Congruent Duoequilateral
Noncongruent Duoequilateral
Base-congruent Equilateralisosceles
Leg-congruent Equilateralisosceles
Noncongruent Equilateralisosceles
Unicongruent Equilateralscalene
Noncongruent Equilateralscalene
Base-congruent Duoisosceles
Leg-congruent Duoisosceles
Transcongruent Duoisosceles
Duotranscongruent Duoisosceles
Congruent Duoisosceles
Noncongruent Duoisosceles
Base-congruent Isoscelesscalene
Leg-congruent Isoscelesscalene
Duocongruent Isoscelesscalene
Noncongruent Isoscelesscalene
Unicongruent Duoscalene
Duocongruent Duoscalene
Congruent Duoscalene
Noncongruent Duoscalene

Here are the formulas:

Congruent Duoequilateral: (a = side length)
SV: (6a^3)/4*sqrt(3)
B: (3a^4)/16

Noncongruent Duoequilateral: (a = side length of first triangle, x = second triangle)
SV: (3xa^2+3ax^2)/4*sqrt(3)
B: (3(ax)^2)/16

BC Equilateralisosceles: (a = leg length of iso, b = side length of equi and base length of iso)
SV: (2ab^2+b^3)/4*sqrt(3)+(3b^2)/2*sqrt(a^2-(b^2)/4)
B: (b^3)/8*sqrt(3a^2-(3b^2)/4)

LC Equilateralisosceles: (a = side length of equi and leg length of iso, b = base length of iso)
SV: (2a^3+ba^2)/4*sqrt(3)+(3ab)/2*sqrt(a^2-(b^2)/4)
B: (ba^2)/8*sqrt(3a^2-(3b^2)/4)

NC Equilateralisosceles: (a = leg length of iso, b = base length of iso, x = side length of equi)
SV: (2ax^2+bx^2)/4*sqrt(3)+(3bx)/2*sqrt(a^2-(b^2)/4)
B: (bx^2)/8*sqrt(3a^2-(3b^2)/4

I'll post the rest of them later...
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Postby Marek14 » Mon Mar 06, 2006 7:45 am

Well, the general formula for surface and volume of duoprism should be:

S = c1*S2 + c2*S1 (where c1,S1 are circumference/area of first polygon, and c2,S2 of second)
V = S1*S2
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Postby Neues Kinder » Sun Jun 18, 2006 2:27 am

This reply is kinda late. I started to lose interest in this site, partially because sometimes it takes every bit of brainpower I have just to even attempt to comprehend what you guys are saying. I did a pretty good job of understanding most of this conversation.

Anyway, I have been thinking about that whole polyprism versus duoprism debate we've had, and I've noticed that I've been somewhat ignorant and stubborn in wanting to keep the name and not noticing that you said it was the official term for those types of prisms. I'll call those duoprisms from now on, and I'll call what I formerly called "duoprisms" something like "rectaprisms" or "tetraprisms".

I think the main reason I was so stubborn was the fact that I personally discovered that you can have a cartesian product of two polygons on my own (but obviously wasn't the first to figure it out). Accepting it as my own discovery, I decided to call them "polyprisms", because they are cartesian products of two polygons, as compared to cartesian products of a polygon and two lines, which I called "duoprisms".

So that I use the official terms and not the terms I just created myself, how do they name the duoprisms? Do they call a cartesian product of two triangles a duotriangular prism, or a tritriangular prism, or something else? Would they call a cartesian product of a pentagon and a heptagon a heptapentagonal prism?
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Postby Marek14 » Sun Jun 18, 2006 7:03 am

Well, a product of two triangles would be just "triangular duoprism". A product of pentagon and hexagon would be "pentagonal/hexagonal duoprism", although I guess that "5-6 duoprism" is more common in writing.

The reason why there's no change to the words in the compound is that you can make duoprism of literaly ANYTHING, regardless on its dimension. So you can have tetrahedral/octahedral duoprism in 6D etc.
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Postby moonlord » Sun Jun 18, 2006 11:35 am

Even if the two components have different dimensionality? Like, pentagon x tetrahedron is a pentagonal-tetrahedral duoprism?

EDIT: "x" is the cartesian product.
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Postby Neues Kinder » Sun Jun 18, 2006 4:02 pm

Yeah, I'd expect you'd name prisms with components of different dimensionality that way. I think that's why you would name a cartesian product of a pentagon and a hexagon a pentagonal-hexagonal prism other than a pentahexagonal prism, because the penta part could be describing a pentahedron.
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Postby Marek14 » Mon Jun 19, 2006 5:34 am

Neues Kinder wrote:Yeah, I'd expect you'd name prisms with components of different dimensionality that way. I think that's why you would name a cartesian product of a pentagon and a hexagon a pentagonal-hexagonal prism other than a pentahexagonal prism, because the penta part could be describing a pentahedron.


Yes, exactly.
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