Base 360

COS Ring

Tim's circular slide rule, seated on one cosine table. Sine is the same ring read 90° earlier. Tangent is one cut division. The voxel pass never calls a transcendental.

  • Ink needle — degree on the table. Drag the bezel.
  • Red pip — log of the absolute reading. The arc is the mantissa.
  • Brass ring — C scale, index seated on that pip.

0° is to the right. Angles run counterclockwise, same as the voxel frame.

Cosine · sealed

0.96592582

e−6 0.965925 · e−4 0.9659

Exact form, then cut: (√6 + √2) / 4

Sine

0.25881904

COS[285] · shift −90°

Tangent

0.26794919

Wide quotient, then cut.

Dividing the 8-digit sine by the 8-digit cosine walks by -1 at the 1e−8 place. Near the axes that walk grows. The face keeps the wide cut.

2.20000000 × cos 15° = 2.12503680

The pass, COS-1

  1. 01Seal COS[k] for k = 0…359. Canonical angles use exact radicals. The rest are a Taylor cosine, then a cut at 1e−8.
  2. 02sin k = COS[k − 90]. One ring, not a second series.
  3. 03tan k is the wide sine÷cosine, then cut. At 90° and 270° the value is quarantined.
  4. 04A voxel heading is a binary search on ray crosses from that ring. The image loop calls no arc-tangent.
  5. 05Shade = luminance × COS[spoke − lamp], cut toward zero. Negative cosine is the dark side of the lamp.
  6. 06Address = shell × 360 + spoke. A full square face is 129600 cells.

CORDIC, the digital slide rule, agrees with this face at 1e−8 on every base angle from 0° through 90°. After the cut, 1 − (sin² + cos²) at 15° is 0.00000002. That remainder is the cut.

Voxel face

Origin at the frame center. 0° points right. Headings come off the ring, then the pass multiplies. No sine, cosine, or arc-tangent runs inside the loop.

Frame · Token
Lambert · sealed cosine

Touch the frame. The cell address is shell × 360 + spoke.

Spokes used
0 / 360
Pass
… ms
Transcendentals
0 in loop
Full face
129600