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.
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?
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.
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.
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.
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.
Users browsing this forum: No registered users and 4 guests