Weird 4D rotatopes

Discussion of shapes with curves and holes in various dimensions.

Weird 4D rotatopes

Postby quickfur » Sun Sep 19, 2004 10:14 pm

I've discovered that the basic 4D rotatopes described by alkaline (cubinder, duocylinder, spherical cylinder) are by no means the only possible "wheels" in 4D. I found out that there are, in fact, an infinite number of 4D shapes that can roll like wheels.

Here's an example: the cylindrical double-cone:
Construction: take a 3D cylinder, and attach 2 cones (or half-cones, for the mathematically pedantic) to its circular ends. Now twist these two cones into the W axis until their apices meet. This produces a 3-member "ring". Now, take a triangle whose apex lies on the origin of the plane, and rotate it around the x-axis so that it forms a triangular torus. This torus can be fitted on the 3-member ring to make a closed 4D object: the cylindrical double-cone. (Provided, of course, that the size of the triangle matches with the length from the base to the apex of the cones.)
Properties: This object has 3 types of rotation. One is on its cylindrical volume which covers a straight line. The other 2 are rotations on the 2 nappes of the cones, which cover the area of 2 circles. It has 2 triangular faces which meet at one vertex, and are rounded off along their edges.

Here's another example: the tri-cylinder:
Construction: take a 3D cylinder and attach two other cylinders on either end. Twist these two cylinders into the W axis until their free circular faces meet. Now construct a triangular torus by rotating around the X axis a 2D triangle with apex facing the origin but displaced along the Y axis by the radius of the cylinders. The resulting torus can be fitted onto the ring of cylinders to form a closed 4D shape.
Properties: this object has 3 faces that it can roll on; the 3 directions it can roll on are 120 degrees apart. It has 2 triangular faces on opposite sides, which are rounded off at their edges.

An even more interesting shape is the tetra-cylinder.
Construction: start with a 3D cylinder, and attach two other identical cylinders to its ends. Rotate these two cylinders into the W axis until they are perpendicular to the first. Now attach a 4th cylinder between the free ends of these two. Now take a 2D diamond (a square rotated 45 degrees), displaced along the Y axis by the radius of the cylinders. Rotating this around the X axis gives a square torus which can be fitted onto the 4 cylinders to form a closed 4D object, the tetracylinder.
Properties: this object can roll in 2 perpendicular directions just like the duocylinder, but it has two round sides per direction (a total of 4 surfaces) that it can roll on. It has 2 square faces which are rounded off at the edges.

As you can probably tell by now, these last 2 objects are really just 4D prisms with cylindrical sides. We can attach N cylinders end-to-end, making a ring in 4D, and "cover" it up into a closed 4D shape by using a torus made by rotating an N-polygon. In this way, you can make prismic "wheels" that rotate in N different directions simultaneously. (Although they will only cover a 2D area.) If you take the limit of N to infinity, you will end up with a spherical prism which can cover a 3D volume by rolling. (This may be the same object as the spherical cylinder, but I'm not sure about that.)

There's also a bizarre object which I call a "bi-cone" for lack of a better name.
Construction: Start with a 3D cone, and stack smaller cones with linearly decreasing height onto it along the W axis until they vanish into a point, forming a 2nd apex an equal distance from the center of the circular base as the 1st apex.
Properties: this object is very strange, in that it has a sharp edge between its two apices, but projects into a 3D cone at 2 different angles. It can also project into 2 cones attached by their bases. Its circular base can actually rotate around the circumference; so it can act like a wheel if balanced correctly. Its sharp edge is perpendicular to this circumference, so it can roll and at the same time have its two apices point in the same direction, like 2 horns. :-)

Now, I'd be darned if I can come up with mathematical formulae for these objects... anyone know where to begin? :oops:
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Oops...

Postby quickfur » Mon Sep 20, 2004 5:00 pm

OK, this will teach me to post before I think... the tetracylinder I described above is, in fact, the same as the cubinder. Also, the toruses I described actually must be rotated through 4D to have the right shape; a 3D torus wouldn't work. :oops:
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Postby wendy » Wed Jan 19, 2005 5:30 am

Here are the names for these and the current lace-prism designations.

> Here's an example: the cylindrical double-cone:
PG= circle-prism peak pyramid xoOoo&oo&#x

> Here's another example: the tri-cylinder:
PG = triangle-circle prism x3o&xOo

> An even more interesting shape is the tetra-cylinder. = cubinder,
PG square-circle prism x4o&xOo

> There's also a bizarre object which I call a "bi-cone"
PG = circle-latron pyramid xooOooo&#x

> duocylinder
PG = bi-circular prism xOo&xOo

> spherical cylinder
PG = glomohedric prism xOoOo&x

Of course there are *many* others

PG = bicircular tegum mOo&mOo.

This is a thing where one covers the circles in the wx and yz planes with a skin.

PG = polygon-circle prism xPo&xOo

A figure formed by the prism-product of a polygon and circle.

PG = glomochorix = xOoOoOo

A four-dimensional sphere

PG = bi-circular ellipsoid xOoOxOo

A glomochorix squashed in a two-dimensional plane, ie an ellipsoid of axies aabb.

Not of course, forgetting the swirlprisms, eg

x5o3o&#s dodecahedral swirl-prism

x4o3x&#s rhombocuboctahedral swirl-prism

and the 120-faced m5m5m&#s
The dream you dream alone is only a dream
the dream we dream together is reality.

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