# SU(3) vs Directional Invariance

1. Jan 18, 2004

### Antonio Lao

The more I read about group theory and SU(3), the stronger is my suspicion that they are very similar to the principle of directional invariance, which is based on the ideas of describing a dynamic one-dimensional cube and its eight properties.

Maybe I am wrong and too engross in my own ideas of imagining a similarity that I seeing the forest from the trees. If there is someone who can shed light on groups and SU(3), I am willing to listen. Thanks.

2. Jan 18, 2004

Staff Emeritus
What do you mean by a "One dimensional cube"?

3. Jan 18, 2004

### ranyart

Quote:dynamic one-dimensional cube and its eight properties.

Surely you mean a 'Planck Cube'?

4. Jan 18, 2004

### Antonio Lao

1-dim cube

The volume of an n-cube width half-side length 1 is

$$V=2^n$$

When n=1 dimension, V=2. But the properties of a cube:
(1) having eight vertices
(2) having six sides
Should still be implied in the 1-cube.

5. Jan 19, 2004

Staff Emeritus
No. The 1-dimensional cube is a line segement with 2 vertices and one edge and no faces.

The general formulas for hypercubes are found at this Mathworld site

6. Jan 19, 2004

### Antonio Lao

Superstring did it?

From my understanding of superstring, it's exactly how the theorists did it. They look very closely at the 1-dim string and found that many dimensions are here compactified. They even have a name for this multi-dim object. It's called a Calabi-Yau shape.

Using abstract math, we can say that a 3-cube can be transform into a 1-cube with 11 edges, 6 vertices, and 6 faces being all compacted but the properties for the "cube" are invariants. Otherwise we can call the transformed object any other name like 1-sphere or 1-tetrahedron or 1-octahedron or 1-dodecahedron, etc. All these other object do satify the Euler polyhedral formula:

V+F-E=2

V is number of vertices, f is number of faces, and E is the number of edges. This formula gives the distinction and the properties for the different objects. That is why they have their different name. It doesn't matter at what dimension we are looking, their properties are invariants.

Last edited: Jan 19, 2004
7. Jan 19, 2004