Exploding tetrahedron  [on:] 2022, paper, glue, ABS 3D-prints, steel, 90 x 90 x 100 cm

Exploding_Tetrahedron

The tetrahedron is the smallest possible body, also called simplex. It is essentially 4-dimensional.

Two perpendicular lines of equal length are not in the same plein. The – of the + is not in the same plein as the | of the +. Connecting the 4 endpoints of the two lines in space with equal length lines form 4 equilateral triangles, the tetrahedron. We get two more perpendicular line pairs with the same spacial distance. They form the 3 dimensions. The tetrahedon has the same shape in all dimensions and is its own dual body. (connecting the midpoints of the surfaces)

The 4 dimension axes go through the 4 vertices and midpoints of the opposing triangle.The 4D simplex or pentachoron unrolled in 3D results in putting a tetrahedon on each side of the tetrahedon. It is like pulling the tetrahedon from each vertex through itself inside out on the triangle side.

The 4 axes devide the central tetrahedron into 4 equal parts, the quarter-tetrahedron. They are each centered on an axes and move through each other outward to appear outside.

Halfing the edges of the tetrahedon reveals the octahedron inside the tetrahedron. The volume of this octahedron is half that of the tetrahedron. So it appears on both sides of the axis.

The axes, 100 cm long, are divided in triangle numbers Δ₉ and Δ₁₀ (45 and 55 cm) .The edge length of the central tetrahedron is the sum of two triangle numbers Δ₂ + Δ₃, 9cm. The visible parts of the axes approach the golden ratio. The endpoints of the longer axis part connected form a tetrahedron. The endpoints of the shorter axis part connected form a smaller tetrahedron mirrored at the center, just like the dual tetrahedron.