The magical 8 of the tesseract (4-D analogue of the cube). A 3D design created in vZome. Use your mouse or touch to interact.

Top: a 3-D perspective (yellow) of the 4-D hypercube (blue). Salvador Dali’s Corpus Hypercubus (cubic cross).

Middle: two larger dual tetrahedra generating an inner octahedron, delineating 8 smaller tetrahedra perfectly filling the 8th (base) cube of the cross at the vertices.

Bottom: smaller central cube (yellow, 3-D perspective), interlocked with 8 cubes perfectly filling the larger cube (blue) generated by oblique (diagonal) projections of the central cube.

References:

Perfect Shapes in Higher Dimensions - Numberphile

A Journey into the 4th Dimension - Perspective - Part 1

Visualizing 4D Geometry - A Journey Into the 4th Dimension - Part 2

Four Dimensional Maths: Things to See and Hear in the Fourth Dimension - with Matt Parker

Generating 4D Polytope Projections with vZome, UPDATED

Quaternions (as rotation operators) 1 2 3 4

The Strange Numbers That Birthed Modern Algebra

PS: Brief notes on philosophical geometry.

Visualize each vertex of a 3-D polyhedron as turning on itself along the “orthogonal” diagonal (orthogonal relative to the higher dimension, projected as diagonal on 3-D), recreating itself fractally at deeper levels consciousness.

Higher dimensions:

  1. X: East-West
  2. Y: North-South
  3. Z: Above-Below
  4. W+: Within-Without or Inward-Outward dimension (scope of consciousness)

Because vertex “movement” (its turning) is implicit in perceiving higher dimensions from a 3-D perspective (human brain), the 3-D concept of time (displacement of the perceiving consciousness) is associated with perceiving higher dimensions. The 3-D observer needs to systematically shift its focus to understand the cross-sections (flipbook) of the 4-D constructs (frames) sequentially presented to the mind’s eye.

An observer who can simultaneously perceive the sequence of frames has transcended the relative lower dimension. The 2-D flatlander can achieve spherical consciousness, as the spherical (3-D) can achieve hypercube (4-D) consciousness, in an ever-expanding spiral of evolution of the scope of consciousness of the perceiving unit.

Among the outer instruments (microscopes and telescopes extending the five physical senses of the material world) emerges an inner (seventh) intuitive sense — an “introscope” — which is to the mind (the sixth sense) what the inner sense of the mind is to the outer five. -JB 2022.10.04