by **wendy** » Thu Dec 21, 2017 2:17 pm

If you take something like a bi-decagon prism, you can create a set of 10 swirls on the surface, by going diagonally in the same direction. These would unwrap to be parallel, and they're equidistant and non-crossing in 4d too. The surface of a glome can be divided into parallel toruses, and for any given torus the rectangle it unfolds to has a diagonal of 2pi r of the glome. The diagonals drawn in a second torus, just under the first set of swirls, would also swirl, and be equidistant from the first. But what happens, is that the two great circles do not lie in a cross-section sphere. These are 'Clifford-parallels' in E3.

If you are familar with the mathematics of complex numbers (y=a+bi, where i² = 1), and with algebraic geometries (ie Y=aX+b, where a is the slope and b is the intercept), the mathematics is straight-forward.

In the real numbers, we can imagine that addition = sliding the number line so 0 becomes a, and any x becomes a+x. Multiplication is done by stretching from 0, so 0 stays at 0, and 1 stretches to a will cause x to stretch to ax. The complex-number addition happens on a plane, where the plane is slid without rotation that (0,0) becomes (a,b), and (x,y) becomes (x+a,y+b). Multiplication is such that we stretch and rotate the plane, so (0,0) becomes (0,0) [that's the pin], and (1,0) becomes (a,b). The general point (x,y) ends up at (ax-by, ay+bx). The algebraic product of (a+ib)(x+iy) is ax+i²by + i(ay+bx ), from which we see that i²by = -by, and hence i²=-1.

Multiplying (a+bi) by (a-bi) gives (a²+b²), this is a point on the line through (1,0). So to divide by a+bi (which rotates a,b onto (1,0)), we multiply by a-bi, and divide by (a²+b²).

In the complex plane, we can introduce a variable 'w', which sets the plane rotating. It is w(t)= 'cos 2pi t + i sin 2pi t'. When t is an integer, a complete rotation has happened. When t=½, all values are multiplied by -1.

Let's now consider the line Y = PX + Q, where X, Y, P,Q are complex numbers, and P and Q are constant for any line.

First, you can see that it is possible to draw a line parallel to S, that passes through any point U. Since we can now find some Y = PX + Q, where where U is some X,Y. Since we know X,Y,P, then Q = Y-PX.

The euclidean rule of only one line through two points holds. But this is now 2-spaces in 4-space. It only holds like here when every point is a swirlybob.

Now let's turn the slope into a sphere. We consider every line that passes through (0,0), that is Y=PX. Now if we multiply X and Y by W(t), our rotating plane, the product is W(t)Y=PW(t)X. But the W(t) are equal in every case, so space in 4d rotates solidly around a point, in non-intersecting 2-spaces.

Now, we can consider another argand diagram with the points P mapped on it. It's a slope diagram. Let's suppose it's in some (x,y,0) space. We put a sphere on the plane, the diameter of which runs from (0,0,0) to (0,0,1). We draw a line from (x,y,0) to (0,0,1) and this is (x,y) mapped on the sphere. The point diametrically opposite is some (-x/r, -y,r), maps onto the completely orthogonal rotation. This is true for every diameter of the sphere.

Every circle around the centre maps onto a 'complex slope' and a radius, the complex slope equates to a direction in 3d. So you can map a three-dimensional scene into a series of circles around the 4d point, such that all points except the centre, become circles at radius r. Put a polyhedron in there, and it becomes a swirlyprism.

Some examples

If you take a dodecahedron, the opposite faces point in opposite directions. We write this as x5o vs o5x. But the 'equator' of a {5,3,3} twelftychoron, has ten of these in a ring. So if you were to imagine going from top to bottom, you rotate 36 degrees, and the bottom pentagon looks like the top one. As you go through the ten decagons, the thing rotates by 36 degrees, each time, so the pentagon appears to rotate as you go around the centre. This is the swirl part.

Each edge of the pentagon is where a third decagon fits in. The pentagon on opposite sides can be found on the 'zigzag decagon' or [petrie polygon] of the circle we just went around. The next circle of dodecahedra have their tops on the middle zigsag, and the pentagons we were crossing appear as an edge on the zigzag. You can follow this around, and it makes a circle of 10 dodecagons, there are five of these directly touching the first.

If we were to join these circles of the dodecahedra together, we would get 12 of them, which with some work give a swirl-dodecahedron.

The swirl tetrahedron comes from the 24-chora. Each face of the tetrahedron is derived from a stack of six octahedra, which rotate 60 degrees each step.

The swirl cube and octahedron, comes from the bi-truncated 24-choron, or octagonny. The octahedron is derived by standing the truncated cubes on their triangle-faces. You get a stack of six of these, and eight stacks make the eight faces of the octahedron. The cube derives from a stacks of eight truncated cubes, each rotated by 45 degrees (so an octagon gives way to a triangle). There are six such stacks.

The icosahedron again derives from our 120choron, but the stacks are dodecahedra on a stick. That is, the side of a initial hexagon, is the dodecahedron and the edge running from the vertex away from the dodecahedron. You get stacks of six of these, and twenty such stacks make the swirl icosahedron.

The relative importance is that this is the major symmetry groups of the 4d polytopes, are swirlybob polyhedra. Only the pentachoron misses out.

Swirlybobs come left and right-handed. Any circle C, gives a completely orthogonal circle G, If C rotates in a given direction, then G rotates one way for a left-hand swirlybob, and the other for a right-handed swirlybob.

So for the examples above, the groups like [3,3,5] have some 72 axies of pentagonal 'wheel' rotation, which gives 144 when direction is considered. This is the product of 12 left dodecahedron-swirls and 12 right dodecahedron-swirls. Its 200 triangle axies come from 400 arrows, being the product of 20 left by 20 right swirl-icosahedra. The 450 digonal rotations, come from 900 rotations with direction, as 30 left- by 30 right- swirl-icosadodeca.

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