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After the discovery i made of the Unity 3 4 0.5 (an object that is the 'simple' addition of a Tetrahedron and a Square Pyramid both with all sides of equal length and joined at both a single point and a single side as well as a single face ( ie. unitarily in all 3 dimensions)) by simply taking a 10x10x10 cube and then rotating the three X,Y and Z planes by 30 degrees to make angles of 60,120 degrees between all 1000 units in place of the 90 degree ones in the cube to make an object of six equal diamond rhombus faces. By halving each face with a line of the same length as the cube sides across the short diamond axis 3 squares appear consisting of paired short axis lines and their opposing face lengths. Each square bisects the object and this results in a pair of Unity 3 4 0.5 objects. (Tetrahedron+Square Pyramid additions). If this is somewhat confusing you can see examples in my photos under the titles starting: Unity 3 4 0.5...

Some very interesting mathematics can be generated using this object and the lengths of its size as they increase in integer values.

These are not limited to volume ratios between the two regular polygon constituents (this limits to 1:2 between Tetrahedron:Sq Pyramid); the approximations of this ratio as the side lengths increase from 0 to infinity; the squares of the side lengths as each object adds a new 'level' or 'depth' particularly for the first 12 'levels'. The connection between square odd numbers and every successive two levels of the Tetrahedron and square even numbers on every even numbered level.

There is still much more work to be done arising from the relationships between the points (that can be seen as circles or spheres of unit size) on comparable levels of each polygon as they increase by a value of 1 from the origin point ( the Top) of each one.

The pattern of volume ratios are determined by the simple equation: Volume ratio V t|p for number (level) N = N+2 : 2N+1

The first 8 levels (starting at level 0)

Produce the following series:

Level    Tet      S.Pyr   Vol Ratio

   0        0          0        2:1*

   1        1          1        2:1*

   2        3          4        4:5

   3        6          9        5:7

   4      10        16        6:9

   5      15        25        7:11

   6      21        36        8:13

   7      28        49        9:15


* these two values do not make sense using the Formula for Volume ratios as volumes do not exist in 1 or 2 dimensions; as with fibonacci sequences though it is important to realise there is a Zero and Unity value for both objects to consider.

Neophyte 2018

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