Laegna · geosis · 8 triangles per square
Seeing the sphere
clearly.
A glass ball unfolds into a square quilt. You paint the quilt. It wraps back — no wrinkles, no lost corners. Eight octahedral triangles, refined in octaves, in exact bijection with the pixels of a square: Tk = 8·4k triangles against Pk = 4k pixels.
The quilt and the ball
Left: the square texture field, every pixel one spherical triangle. Right: the same field wrapped back. Hover the quilt — the face and the cell index under your cursor are the coordinates of a patch on the ball.
The octave ladder
Each step up is one octave of detail: four times the triangles, four times the pixels, half the angular spacing. Resolution grows harmonically, like doubling a frequency.
Refinement commutes with projection
Refinement splits a triangle at its midpoints. Projection preserves midpoints and edges. So sub-triangles map to sub-triangles, and it does not matter whether you refine first and project after, or the reverse. The mapping is structurally exact at every octave.
Russian-doll spheres
Angular resolution does not care about radius. Physical smoothness does. The same mesh on a marble and on a planet: identical angles, wildly different patches. This is why 256×256 looks like glass on a marble and like gravel on a world.
Four bodies, one field
Each body is written into the same octahedral square and re-wrapped. Nothing here is a lat-long map: the poles carry no singularity, because there are no poles — only eight equal triangles.
oceans, continents, ice caps
ochre dust, maria, polar frost
grey regolith, dense cratering
bulge glow, dust lane, star field
Dimensionality and complexity loss
Element count is a squared law in the side length and a fourth-power law in the octave; angular precision is only linear in the halving. Every honest algorithm chooses a finite k and lives inside that gap — but because the refinement is uniform and symmetric, the error is predictable, and interpolation between octaves is well-defined.
Interpolation
Between k and k+1 nothing is invented: each parent cell contains exactly four children, so a value at octave k is the mean of its four descendants. Refinement is a lossless split; coarsening is an average. The ladder is reversible.
Radius independence
The mesh lives in angle, not in metres. Choose k for precision and r for scale separately — two knobs that never interfere. This is what makes one texture serve a marble and a galaxy.
GPU shape
The whole projection is s = |x|+|y|+|z| and a fold. No trigonometry, no branch-heavy pole handling — a vertex shader can carry it, which is why this page runs the same math on the CPU quilt and on the GPU ball.
Geosis
Strip away the proofs and a human idea remains: take something curved and complex, flatten it into something simple, add detail in harmonic steps, and wrap it back without losing anything.
Spirit understands. To be spiritual, here, is to simplify until a thing can be held — to cognite, to build a model whose symmetries are the symmetries of the matter it describes. Geometry is not decoration on top of mathematics; it is the syntax in which mathematics becomes visible, and therefore livable.
laGEOsis is Laegna geosis: form, disposition and colour treated as first-class carriers of meaning. A better geometry is a better sentence. The octahedral quilt is such a sentence — it says sphere in square, harmonically, with no seam to apologise for and no pole to special-case.
What is drawn on this page is drawn twice by the same functions: once into pixels, once into vertices. That doubling is the point. A model that survives translation between representations is a model that has understood something.