# Message #1480

From: Andrey <andreyastrelin@yahoo.com>

Subject: Re: [MC4D] Twists description

Date: Thu, 03 Mar 2011 05:28:35 -0000

Hi Andy,

"Movements:" give us description of the symmetry group of the puzzle body. Vectors from this set define reflections around their orthogonal hyperplanes. Movements are used to define the set of axes and their order, but vectors fr0om this set don’t have to be vectors of multi-axes, or moving planes or so on (but usually they are).

"Vector:" is a part of description of the cutting hyperplane. It is orthogonal to a set of parallel hyperplanes, and it always belongs to multi-axes of all twists of all layers cutting by these hyperplanes.

"Planes:" it’s desription of moving planes of the layer orthogonal to "Vector". Each moving plane is defined by a pair of vectors. Both of them are perpendicular to "Vector". Angle between vectors is a half of the minimal twist angle (so we are able to define 180-deg twists by selecting of two perpendicular vectors).

"Cuts:" distances from the center to cutting hyperplanes (in the direction of "Vector"). For the symmetrical bodies set of distances is usually symmetrical, but for simplex and some other bodies like (4,5)-duoprisms positive and negative directions of the axis are different.

And yes, multi-axis is (D-2)-dimensional space.

Andrey

— In 4D_Cubing@yahoogroups.com, Andrew James Gould <agould@…> wrote:

>

> Hi Andrey,

> I forgot both are planes of rotation in 4D. I’ll used the words ‘fixed’ and ‘moving.’

>

> Does ‘Movement:’ = R1 vectors and

> ‘Vector:’ = R vector

> from your previous email? They make the fixed multi-axis [R,R1]?

>

> Consider an N-dimensional simplex (N>2). For any 120-degree rotation that can do a 3-cycle on 3 vertices (and the 2D triangle between them), the moving plane of rotation is determined as well as the fixed multi-axis. Orient the simplex so that the moving plane of rotation is an x-y rotation and move the simplex so that the center of the simplex is at the origin. Then since the moving plane of rotation is orthogonal to the multi-axis, (1) the multi-axis is the restricted set x=0 and y=0. Also, since the remaining fixed vertices are equidistant from the moving verticies, (2) the x and y coordinates of the fixed vertices are restricted to 0. (1) and (2) imply that a multi-axis for such a 120-degree rotation contains all fixed vertices. (It also contains the center of the moving triangle.) That help?

>

> I assumed your multi-axis is everything spanned by whatever axes you list in it.

>

> –

> Andy

>