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What Is An Adi Polytope?

written by Adi Cox 1st April 2015_____________________________________________________________________

There is a video on youtube that accompanies this paper:

https://www.youtube.com/watch?v=qCAB1HPbC-Y

Introduction____________

To find out what an Adi polytope is we need to understand what a measure and a simplex is. This is because an Adi polytope is a measure with some right simplexes taken away from it.

There is an Adi polytope in every dimension that is 2 or above. This is because there are no right simplexes in less than 2 dimensional space. It does get confusing because the 2nd dimensional Adi Polytope is a one dimensional line.

An explination of the notation used:

xyz xyz xyz xyz000 {001, 010, 100}

point 000 is connected to points 001, 010 and 100.

where xyz are the axis in space.

What Is A Measure?__________________

There is a measure in every dimension. We can use binary to find measures in any dimension:

0th dimension measure is the point:

0

1st dimension measure is the line:

0 1

2nd dimension measure is the square:

00 {10, 01} 01 {11, 00} 10 {00, 11} 11 {01, 10}

3rd dimension measure is the cube:

000 {100, 010, 001} 001 {101, 011, 000} 010 {110, 000, 011} 011 {111, 001, 010} 100 {000, 110, 101} 101 {001, 111, 100} 110 {010, 100, 111} 111 {011, 101, 110}

4th dimension measure is the tesseract:

0000 {1000, 0100, 0010, 0001} 0001 {1001, 0101, 0011, 0000} 0010 {1010, 0110, 0000, 0011} 0011 {1011, 0111, 0001, 0010} 0100 {1000, 0000, 0110, 0101} 0101 {1101, 0001, 0111, 0100} 0110 {1110, 0010, 0100, 0111} 0111 {1111, 0011, 0101, 0110} 1000 {0000, 1100, 1010, 1001} 1001 {0001, 1101, 1011, 1000} 1010 {0010, 1110, 1000, 1011} 1011 {0011, 1111, 1001, 1010} 1100 {0100, 1000, 1110, 1101} 1101 {0101, 1001, 1111, 1100} 1110 {0110, 1010, 1100, 1111} 1111 {0111, 1011, 1101, 1110}

What Is A Simplex?__________________

A simplex has all points connected. In n dimensions the simplex in that dimension has n+1 points. Using binary we can look at the regular simplex and the right simplex:

0th dimension simplex is the point:

0

1st dimension simplex is the line:

0 1

2nd dimension simplex is:

00 {01, 10} 01 {10, 00} 10 {00, 01}

Above is a right simplex because there is a right angle at point 00.

3rd dimension simplex is:

000 {001, 010, 100} 001 {000, 010, 100} 010 {000, 001, 100} 100 {000, 001, 010}

Above is a right simplex because there is a right angle at point 000.

100 {010, 001, 111} 010 {100, 001, 111} 001 {010, 001, 111} 111 {100, 010, 001}

Above is a regular simplex because all side lengths are equal

4th dimension simplex is:

0000 {0001, 0010, 0100, 1000} 0001 {0000, 0010, 0100, 1000} 0010 {0000, 0001, 0100, 1000} 0100 {0000, 0001, 0010, 1000} 1000 {0000, 0010, 0100, 0001}

Above is a right simplex because there is a right angle at point 000.

1000 {0100, 0010, 0001, 1111} 0100 {1000, 0010, 0001, 1111} 0010 {1000, 0100, 0001, 1111} 0001 {1000, 0010, 0001, 1111} 1111 {1000, 0100, 0010, 0001}

Above is a regular simplex because all side lengths are equal

So what is an Adi Polytope?___________________________

An Adi Polytope is an n dimensional measure with 2^(n-1) right simplexes taken away from it.

E.g. The 2nd dimension is a trivial example:

Start with a square and taking away two right angle triangles we get just the straight line where the hypotinuse of both right angle triangles meet. So in the second dimension the Adi Polytope is just a straight line.

2nd dimension Adi Polytope is:

2nd dimension measure minus 2nd dimension right simplexes 1 and 2

2nd dimension measure:

00 {10, 01} 01 {11, 00} 10 {00, 11} 11 {01, 10}

2nd dimension right simplex 1:

00 {01, 10} 01 {10, 00} 10 {00, 01}

2nd dimension right simplex 2:

11 {01, 10} 01 {10, 11} 10 {11, 01}

So when we take away both of these right simplexes we are left with nothing, everything is cancelled out in the second dimension, but we get a 1 dimensional line. This line is from the points 01 10 and this is the first Adi polytope.

3rd dimension Adi Polytope is:

3rd dimension measure minus 3rd dimension simplexes 1,2,3 and 4

3rd dimension measure:

000 {100, 010, 001} 001 {101, 011, 000} 010 {110, 000, 011} 011 {111, 001, 010} 100 {000, 110, 101} 101 {001, 111, 100} 110 {010, 100, 111} 111 {011, 101, 110}

3rd dimension right simplex 1 is:

000 {001, 010, 100} 001 {000, 010, 100} 010 {000, 001, 100} 100 {000, 001, 010}

3rd dimension right simplex 2 is:

100 {010, 110, 111} 010 {100, 110, 111} 011 {100, 010, 111} 111 {100, 010, 110}

3rd dimension right simplex 3 is:

100 {001, 101, 111} 001 {100, 101, 111} 101 {100, 001, 111} 111 {100, 001, 101}

3rd dimension right simplex 4 is:

010 {001, 011, 111} 001 {010, 011, 111} 011 {010, 001, 111} 111 {010, 001, 011}

So when we take away these four 3 dimensional right simplexes from the 3 dimensional measure we get a regular simplex. In the language of the 3rd dimension. We take away four right tetrahedrons from a cube and we are left with a regular tetrahedron as the Adi Polytope of the 3rd dimension.

Regular 3rd dimension simplex = Regular Tetrahedron = Adi 3 Polytope

100 {010, 001, 111} 010 {100, 001, 111} 001 {100, 010, 111} 111 {100, 010, 001}

If we do the same as above in 4th and 5th dimensional space we get the following Adi Polytopes respectively: