by **Marek14** » Fri Mar 03, 2006 7:30 am

Let's see...

Duocylinder's equation is max(x^2+y^2, z^2+w^2) = 1. We can rotate it in six coordinate planes, but obviously rotating in xy or zw does nothing. Let's try xz:

x' = x cos alpha - z sin alpha

z' = x sin alpha + z cos alpha

For alpha = 45 degrees, we have

x' = sqrt(2)/2 * (x - z)

z' = sqrt(2)/2 * (x + z)

So the new equations are:

max((x-z)^2/2 + y^2,(x+z)^2/2 + w^2) = 1

max(x^2/2 + y^2 + z^2/2 - xz,x^2/2 + z^2/2 + w^2 + xz) = 1

Let's put x=0 here:

max(y^2 + z^2/2,z^2/2 + w^2) = 1

simplifying to

max(y^2,w^2) = 1-z^2/2

This 3D object is a sort of dome (one of my extended rotatopes) which is made of square slices with square of edge 2 in yw plane and reducing to point in z=sqrt(2) and z=-sqrt(2). This is what you get slicing through hyperplane yzw. The slices of this object are: square max(y^2,w^2) = 1, and two ellipses, y^2+z^2/2 = 1 and z^2/2 + w^2 = 1.

Let's now try to eliminate z from the original equation:

max(x^2/2 + y^2 + z^2/2 - xz,x^2/2 + z^2/2 + w^2 + xz) = 1

z=0

max(x^2/2+y^2,x^2/2+w^2)=1

max(y^2,w^2)=1-x^2/2

giving exact analogue, which was to be expected.

I didn't try slicing through y or w (the formulas become more complex there), but I would expect that it will still be cylinder, just rotated (after all, we rotated in other directions).

Would the same method work for other rotatopes rotated through 45 degrees? Since equations of the majority of them only contain squares, we have transformations

x1'^2 = (x1 - x2)^2/2

x2'^2 = (x1 + x2)^2/2

If we put x2=0 here, we get equations

x1'^2 = x1^2/2

x2'^2 = x1^2/2

and with this replacement in original equation we should get the equation of x1-slive after rotating 45 degrees in x1x2 plane.

Let's try it...

Square: max(x^2,y^2) = 1. Rotating through xy.

max(x^2/2,x^2/2) = 1 => x^2=2 => x=sqrt(2), x=-sqrt(2)

We get two points of distance 2 sqrt(2), which is correct.

Cube: max(x^2,y^2,z^2) = 1. Rotating through xy.

max(x^2/2,x^2/2,z^2) = 1 => max(x^2/2,z^2) = 1 => rectangle with boundaries z=1,z=-1,x=sqrt(2),x=-sqrt(2)

Cylinder: max(x^2+y^2,z^2) = 1. Rotating through xz.

max(x^2/2+y^2,x^2/2) = 1 => x^2/2 + y^2 = 1. Ellipse.

Tesseract: max(x^2,y^2,z^2,w^2) = 1. Rotating through xy.

max(x^2/2,x^2/2,z^2,w^2) = 1 => max(x^2/2,z^2,w^2) = 1. Cuboid of edges 2, 2, and 2 sqrt(2)

Cubinder: max(x^2+y^2,z^2,w^2) = 1. Rotating through xz.

max(x^2/2+y^2,x^2/2,w^2) = 1 => max(x^2/2+y^2,w^2) = 1. Elliptical cylinder with height 2 and base ellipse 2 x 2 sqrt(2)

Cubinder: max(x^2+y^2,z^2,w^2) = 1. Rotating through zw.

max(x^2+y^2,z^2/2,z^2/2) = 1 => max(x^2+y^2,z^2/2) = 1. Cylinder of height 2 sqrt(2).

This is sufficient to make a conjecture, which provides the following way to construct the slice:

1. Represent the rotatope in graph form (for normal rotatopes, the graph will be comprised of several complete graphs Kn). Write number 1 to each node (it represents half the diameter of the rotatope along that particular coordinate axis).

2. Pick any two nodes.

3. Identify both nodes. New node will have all the edges the previous nodes had.

4. Mark the new node with sqrt(2) if the original nodes weren't connected or with 1 if they were.

So, shortening sqrt(2) to just "2" for square (1) (1) we get (2).

For circle (1)-(1) we get (1) - the same thing as if we just omitted one node.

For cube (1) (1) (1) we get rectangle (2) (1)

For cylinder (1)-(1) (1) we get either square (1) (1) or ellipse (2)-(1)

For sphere (1)-(1)-(1)- we get circle (1)-(1). In summary, rotating in circular plane has no effect on slice in 3D (it's not the case in higher dimensions!)

(For dome (1)-(1)-(1) we get either circle (1)-(1) (if we rotate it in one of its circular planes, the cut will still be circle), or ellipse (2)-(1))

In 4 dimensions:

For tesseract (1) (1) (1) (1) we get cuboid (2) (1) (1)

For cubinder (1)-(1) (1) (1) we get either:

- cube (1) (1) (1)

- elliptic cylinder (2)-(1) (1)

- or cylinder (1)-(1) (2)

For duocylinder (1)-(1) (1)-(1) we get either:

- cylinder (1)-(1) (1)

- or dome (1)-(2)-(1)

So far it matches. Let's verify it for spherinder:

For spherinder (1)-(1)-(1)- (1) we get either:

- cylinder (1)-(1) (1)

- or ellipsoid (2)-(1)-(1)-

Let's look at the equations:

Spherinder: max(x^2+y^2+z^2,w^2) = 1. Rotating through xw.

max(x^2/2+y^2+z^2,x^2/2) = 1 => max(x^2/2+y^2+z^2) = 1. Yep, it's ellipsoid.

For glome, we always get sphere.

How about other 4D extended rotatopes (a.k.a. graphotopes)?

Dominder (y)-(x)-(z) (w): Rotating through xy or xz leads to ordinary cylinder. Rotating through xw leads to dome (1)-(2)-(1), rotating through yz to elliptic cylinder (2)-(1) (1), rotating through yw or zw to dome (2)-(1)-(1)

Tridome (y)-(x*)-(z) *-(w): Rotating through xy, xz, or xw leads to ordinary dome. Rotating through yz, yw, or zw leads to dome (2)-(1)-(1)

Longdome (z)-(x)-(y)-(w): Rotating through xy leads to dome (1)-(1)-(1)! This is significant because ordinary slice through either x or y without rotation leads to just a cylinder. Rotating through xz or yw leads to ordinary dome. Rotating through xw or yz leads to dome (1)-(2)-(1), and rotating through zw leads to ellipsoid (2)-(1)-(1)-*

Spheridome (w)-(x*)-(y)-(z*)-: Rotating through xy or xz leads to ordinary dome. Rotating through xw leads to ordinary sphere. Rotating through yz leads to ordinary dome. Rotating through yw or zw leads to ellipsoid (2)-(1)-(1)-.

Cyclodome (x)-(y)-(w)-(z)-: Rotating through xy,xz,yw, or zw, leads to sphere. Rotating through xw or yz leads to dome (1)-(2)-(1)

Semiglome: three possible results are sphere, dome, or ellipsoid.