Principles of Nature: towards a new visual language
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Triangular units of area:


Figure 65 (figure numbers correspond to the book numbering) demonstrates that successively larger triangles generated by p^{2} configurations of these smaller scalene units are geometrically similar to the original scalene triangular unit from which they were constructed (i.e. their angles are equal, and their sides are in the same ratios, since all sides of the unit triangle ABC are successively multiplied by 2, 3, 4, etc to yield the triangles above). This means in effect, that the p^{2} sidesymmetry of the square and equilateral triangle is maintained here, but with a twist. For example, in the final 16part triangular arrangement above (representing the 4^{2}configuration ), the overall triangular shape can be seen to have all sides divided into four units of length (as in a 4 x 4 square), but now (and here's the twist) the unit of length has also become relative rather than absolute. The unitsoflength on each side are sidespecific (or in other words, siderelative). The asymmetry of the scalene triangle has imposed the need to 'match side with unittriangle side' in order to maintain this squarelike property and composition. In other words four unitsoflength on one side of the triangle do not, in an absolute sense, equal four unitsoflength on another side of the same triangle, only in a relative sense, relative to the respective sides of the scalenetriangularunit as given above. The numbers are equal, but the lengths are relative.
Therefore if the scalene triangular units of length are viewed in this relative way, they too exhibit the characteristics and symmetries of the ‘square numbers’, including the ability to tile the plane without gaps.
The area of a scalene triangle in selfsimilar scalene units is thus also p^{2}, where p is the siderelative length stated in terms (and multiples) of the length as given by the scalene triangle unit’s respective side.
A scalene triangle (coloured blue in the diagram below) in the Euclidean plane can be represented as a crosssection through a cube.
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