What is this object called?

[Teragon]

It's created easily but it has some weird properties. Take a cone and rotate it around its base so that the top traces out a circle in zw. The resulting object is build similar to a duocylinder, but with dicones as sections instead of cylinders. One can also see it as a disk that is tapered while extending radially outward in zw to form another circle as its edge. The resulting shape consists of only one face coiling around two 90°-edges in the shape of orthogonal circles. It has two rotational symmetries - one in the xy-plane, one in the zw-plane.

It's rolling behaviour is for sure the most complex among simple 4D objects. It would be an intersting object just to push and watch moving.

- When its lying on its face, one point of each edge is touching the ground.
- It can roll in two directions (and linear combinations of them). When it's rolling in one of them, the point one edge touches the ground stays the same while the other edge roles like an inclined wheel, tracing out a circle.
- Even though it has only two degrees of motion it can trace out the whole 3-plane. This is a feature of the trajectories in the two main directions being curved - if it rolls around one circle, the plane of the other circle is rotated - it traces out a circle on a circle on a circle and so on.
- I have no idea what a movement off the main rolling directions looks like. The trajectories might look chaotic or form nice open or closed patterns. Would be interesting to simulate them with the computer (anyone interested? ).
- Moving by a distance smaller than the object itself in the third direction (where it can't go directly) is not an easy task.

[wendy]

It's called a bicircular tegum.

[Teragon]

Thanks. Where does the word "tegum" come from?

[wendy]

I invented it. It's a kind of product like 'prism product', that covers the various bases at perpendicular.

I've known of the bicircular tegum since the late seventies, it's one mean test for 4d visualisation!

[ICN5D]

That's exactly what I was thinking, too : a bi-circular tegum. I animated the slices of it, here :

http://hi.gher.space/forum/viewtopic.php?p=23437#p23437

[Prashantkrishnan]

- When its lying on its face, one point of each edge is touching the ground.
- It can roll in two directions (and linear combinations of them). When it's rolling in one of them, the point one edge touches the ground stays the same while the other edge roles like an inclined wheel, tracing out a circle.
- Even though it has only two degrees of motion it can trace out the whole 3-plane. This is a feature of the trajectories in the two main directions being curved - if it rolls around one circle, the plane of the other circle is rotated - it traces out a circle on a circle on a circle and so on.
- I have no idea what a movement off the main rolling directions looks like. The trajectories might look chaotic or form nice open or closed patterns. Would be interesting to simulate them with the computer (anyone interested? ).
- Moving by a distance smaller than the object itself in the third direction (where it can't go directly) is not an easy task.

When it lies on an edge, one point of that edge touches the ground, and the other edge is parallel to the ground. In this position, it can roll in the direction given by the edge which touches the ground. When it does so, it undergoes straight line motion.
When it lies on its cell, one point of each edge is touching the ground. In this position, one line segment of the cell is touching the ground. When it rolls on one edge, this line traces out a cone.

Suppose the vertical is represented by z-axis, and the line segment lying on the ground is represented by y-axis. When the bicircular tegum lies on its cell, one edge would be in a plane given by the linear combination of the xy and the xz planes, in which case, the tegum can roll in the x-direction along this edge, tracing a circle in the xy plane. The other edge would be in a plane given by the linear combination of the wy and the wz planes. Along this edge, the tegum can roll in the w-direction, tracing out a circle in the wy plane. I suppose this is what you mean when you say that it has two degrees of motion, but it can trace out realmspace.

What remains for me to understand is what you call the third direction. Is it the z-direction or the y-direction?

If it is the z- direction, you can just lift the tegum. Otherwise, it will have to slide along its line of contact with the ground. It definitely is not an easy task, but we can obviously make it slide. I don't see what complications arise from the distance being smaller than the object itself.

[quickfur]

Rolling on an edge is unstable; the object will likely fall down on one facet. It's unstable in 3D, and even more unstable in 4D because of another dimension of directions in which it can fall.

The bicircular tegum has a single surface, in fact; a single toroidal 3-manifold spans the space between its two circular edges. I'm unsure how it would roll, though. Intuition suggests rolling would cover a 2D area, but I suspect the actual motion would be quite complex, maybe even chaotic, because of the angle of the surface w.r.t. the ground. It wouldn't roll in a straight line, but probably in a looping or spiralling motion. Just a wild guess.

[wendy]

The actual construct of the bicircular tegum, is a tetrahedron, whose two opposite sides have been rendered as circles, rather like a cone is a tringle bent.

The thing standing on the table, would stand on a line drawn between the two axial circles. One can then rotate either or both of the axial circles, which means the figure would rotate varingly stationary around a point, or in a straight path, as both parts rotate at the same speed. It is, as quickfur notes, a very frustrating figure to visualise rolling on a table.