N dimensional Simplex

Zero D

Nothing inside 0D Point

1 D

0D Point inside 1D line

2 D

1D line inside 2D triangle

3D

 2D triangle inside
3D pyramid

4D

 Task

Table the numbers of dimensions, vertices, and faces to find patterns.

Make up a sets of rules to find the number of vertices and faces for a 4D, 5D and 10D simplex.

Graham Partis

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Today's Notes

N dimensional (nD) triangles referred to in Benji's TOE are elsewhere called simplexes as they are the perhaps the simplest (straight edge) shapes that can be made from the least straight edges. From memory, they are produced by adding another vertex in the new dimension.
 
                                                                                                                       Draw the next 2 shapes

Spheres are produced by spinning the old shape through the new dimension and are in many ways simpler than simplexes.

N dimensional  cubes are constructed by duplicating the whole shape in an extra dimension and joining the two shapes with right angled connectors. These are arguably more complex shapes but (as far as I can tell) can fit together in a grid pattern.

Researching interlocking shapes such as maximal straight edged interlocking shapes such as 2D hexagons may be another way of generating a class of nD shapes.

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