10d • General discussion
The Geometry Atlas
After the Space-Time Cones, I'd like to share the bigger project behind it, the place where all of this geometry lives.
🌌 Have you ever seen the shadow of a four-dimensional shape?
We can't see the 4th dimension directly. But we can see its shadows. A 3D cube held under a lamp casts a flat 2D shadow on the table. In the same way, a 4D shape casts a 3D shadow into our world. Flatten that again and you get a 2D pattern.
That's what the Geometry Atlas is: a map of those shadows.
What's inside
Nine interactive models, each one you can rotate, zoom and look inside:
🔷 The six regular 4D polytopes, the 4D "cousins" of the Platonic solids:
• the 5-cell, the 4D tetrahedron and the simplest 4D shape
• the Tesseract, the 4D cube
• the 16-cell, the 4D octahedron
• the 24-cell, a shape that exists only in four dimensions, with no 3D equivalent
• the 600-cell and the 120-cell, the largest and most complex of all, built entirely on the golden ratio
🔷 The 600-cell and 120-cell merged into one dual pair. A slider lets you grow one inside the other and watch the exact moment they lock together: when the 120-cell is φ times the 600-cell.
🔷 Nested Platonic solids and nested dual shells: tetrahedron, cube, octahedron, icosahedron and dodecahedron sitting inside one another, layer by layer, each touching the next at exact points.
How you can explore them
🔄 Rotate in 3D, and on some models even rotate through the 4th dimension and watch the shadow turn itself inside out.
👁️ Look along the symmetry axes: vertex-first, cell-first, edge-first, face on, and 2-, 3-, 4- and 5-fold views.
🎨 Switch each shell on and off to see how the layers build up from the centre.
📏 Read every rod length in millimetres for the size you choose. Each model is also a parts list for building it physically from copper rods and hubs.
The orthographic views: where it gets beautiful
When you look straight down a symmetry axis, the 3D shadow flattens into an exact 2D pattern: rings within rings, stars within stars. The 600-cell even produces a 30-fold mandala.
And this is what moved me most. These patterns look like the drawings we work with in our Alchemical Manual: the same forms seen from different angles, rings nested inside rings, and centres radiating outward.
Each ring sits at an exact ratio to the next (φ in the golden family, √2 and √3 in the cube family). Moving inward and outward through these shells feels like moving through octaves, with overtones above and undertones below, all spaced by the same proportions. In the Atlas you're not just reading about it. You can see it, turn it, and peel it layer by layer.
How we built it
Nothing in the Atlas is drawn by hand. Every model is generated from exact mathematical coordinates. The 600-cell's 120 corners come straight from golden-ratio numbers, and every edge, cell and shadow is calculated from them. The 30-fold views are projections onto the special plane where the 4D symmetry shows itself most fully (the Coxeter plane). Every count and alignment on the page was checked numerically before publishing, using the classic reference work by H.S.M. Coxeter.
Why we built it
🔸 To see what can't be seen. The 4th dimension becomes something you can hold in your mind and in your hands.
🔸 To build. Our prototypes, such as the Space-Time Cones, the Merkaba and the copper frames, need exact proportions. The Atlas gives the real measurements.
🔸 To connect the principles to the geometry. The Alchemical Manual describes the laws, and the Atlas shows the shapes those laws produce.
Companion studies inside the Atlas
📘 The Five-Fold Ladder: start from a circle divided into five, discover the golden ratio it forces, and follow it all the way up to the 600-cell, where the ladder ends.
📘 Space-Time Cones: φ or √2?: the cone study I shared last time, with the 3D model and Blender files.
No maths background needed. Just open it, click, rotate and explore. 🙏
💛 Which shape speaks to you the most? And which view reminds you of a drawing from the manual? Share a screenshot in the comments.
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The Geometry Atlas
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