Cylindrone or duo-cone?

[Eric B]

On quickfur's site; http://eusebeia.dyndns.org/~hsteoh/4d/cylindrone.html I see:

The Cylindrical Cone
The cylindrical cone, or cylindrone for short, is constructed by tapering a 3D cone along the W-axis.

I had a question about this, and PM'd him, but he didn't seem to see it, and he has no e-mail anywhere. So I figured I could ask publicly, and he or others could answer.

Do you mean "tapering a 3D cylinder along the w-axis"? Because that is what the object shown and discussed there is. I had been thinking about what a conic hypercone (cone tapered along the W-axis) would be like, and I figured the circular base extruded and shrank down to a point would generate another cone. So you would have two cones sharing a common circular base, and ending in two separate apices, joined by a ridge (the extruded original apex, and all enveloped by a triangular torus). I wondered why I didn't see this on your site anywhere, and then I realized that the object you have here is the opposite of this: 2 cones sharing a common apex, with separate circular bases joined by a cylinder. This would be a cylinder extruded and tapered down to a point, not a cone. So this is truly a cylindrone, but what is described is not cylindrical at all, but rather duo-conic: a single circular base in the x and y axes, tapered to points along both the z and w axes.

Anyway, this stuff is fascinating; and I have been studying your page to know more about the duocycles. Before; all my knowledge of the 4th dimension was confined to the simpler "regular" objects.

Eric B

[wendy]

the assorted names like duocylinder etc are not truely reflective of the actual constructions.

The figure described is a cylinder-prism, or by my notation, oxOoo&ox&#t.

This is a (( circle * line ) prism * point pyramid) . I have not formerly considered a gradation for nested products as yet, but the reccomendation that exists is to create "open bracket" operators that the product closes, eg

[[ circle line ]prism point ]pyramid]


W

[Marek14]

I was looking at this yesterday, trying to find how many 4D thingies can be gotten from applying of prism, tegum, and pyramid products on lines, points and various spheres. I condensed the notation a bit, so this would be [2x1]^0
where 2 is circle, 1 line, 0 point, x prism and ^ pyramid (I chose ^ because pyramids tend to be pointy )
This allowed me to appreciate the relationship between tegums and pyramids (since tegums have pyramids as their elements, and can be basically understood as multiple pyramids glued together).
Also, I was a bit puzzled by square-pyramid + line tegum. It seems that when you cut a hexadecachoron in half, you can do it this way or another (which produces octahedral pyramid)?

Where my imagination fails so far, is duocircle tegum. More specifically it's surface which should be S1^S1 pyramid, but I'm having trouble imagining this...

[wendy]

Pretty nifty, eh.

The surface of the mOo&mOo (bi-circular tegum), is actially a tetrahedron, or S1.S1 pyramid, as you note.

A line can be made a circle by connecting the ends,

Take a tetrahedron, and mark two opposite edges as the edges of this tegum. You open one side to a circle, to get a bi-cone. The two edges join, and you can remove the pair of faces that join.

Now do the same with the other pair. In effect, you get two linked circles in 3d, but they are orthogonal in 4d. (none the same they are still linked: imagine trying to make one disappear without comming within a degree of the other. you just can't do it.).

This makes a circle.

Cutting an 16choron in half gives room for a pyramid / tegum mix. zb

** pyramid #* tegum *# prism ## comb or torus

(( line ** pt ) #* line #* line ) works

(( line #* line #* line) ** pt ) works

The first only works if the thing is edge-tangent, i suppose, because it correctly requires the faces to merge, which is not really what #* implies. So i'd guess only þe second one works.

W

[wendy]

The page clearly describes a cylinder-point-pyramid, (which is what the cone is referring to), and i imagine the cone-tapering occurs over w (which would by necessity would be the height of þe prism.

[quickfur]

On quickfur's site; http://eusebeia.dyndns.org/~hsteoh/4d/cylindrone.html I see:

The Cylindrical Cone
The cylindrical cone, or cylindrone for short, is constructed by tapering a 3D cone along the W-axis.

I had a question about this, and PM'd him, but he didn't seem to see it, and he has no e-mail anywhere. So I figured I could ask publicly, and he or others could answer.

Do you mean "tapering a 3D cylinder along the w-axis"? Because that is what the object shown and discussed there is.

I've replied Eric's message, but just for the record: this was a typo on my website. I've just corrected it. Thanks, Eric, for the correction! :-)

[Eric B]

Thatks for responding! I couldn't wait to see your reply.
Here is the duo-cone (for some reason the image didn't come up in the PM)

[wendy]

the duocone is nothing more than a line-circle pyramid.