The Catharturgy Corpus · Christopher E. Etter

The Perpendicular Frame

The limits are not on the shapes within space but in the frame that space is. A dimension is a perpendicular axis — a two-fold, positive and negative — so the right angle has a prime, and the prime is two. The frame of space is built by the primes.

Two — the prime of the right angle
each new perpendicular axis doubles the right-angled form

The hypercube has 2d vertices — one doubling for every perpendicular axis raised. Perpendicularity is the two-fold, and the frame of each dimension is the pure accumulation of 2. No other prime ever enters the right-angled form.

↓ every dimension carries three numbers · hover any row ↓
① The Frame
2d
Hypercube vertices — orthogonality made solid. Pure powers of two: the right angle accumulated once per axis.
② The Skeleton
2·d
Cross-polytope vertices — the ± of each axis. The dimension-number itself appears, so each dimension is prime or composite in its frame.
③ The Ceiling
5 → 7 → 11 → 13
Largest prime in the regular forms. Holds at 5 through the fifth dimension, then rises by threshold as primes enter space.
Dim
① Frame — 2d
② Skeleton — 2·d
③ Ceiling
1D
21 = 2 hypercube vertices · the right-angled frame
2·1 = 2 axis-skeleton composite
2 largest prime in regular forms
2D
22 = 4 hypercube vertices · the right-angled frame
2·2 = 4 axis-skeleton prime dimension
3 largest prime in regular forms
3D
23 = 8 hypercube vertices · the right-angled frame
2·3 = 6 axis-skeleton prime dimension
3 largest prime in regular forms
4D
24 = 16 hypercube vertices · the right-angled frame
2·4 = 8 axis-skeleton composite
5 largest prime in regular forms
5D
25 = 32 hypercube vertices · the right-angled frame
2·5 = 10 axis-skeleton prime dimension
5 largest prime in regular forms
6D
26 = 64 hypercube vertices · the right-angled frame
2·6 = 12 axis-skeleton composite
77 enters largest prime in regular forms
7D
27 = 128 hypercube vertices · the right-angled frame
2·7 = 14 axis-skeleton prime dimension
7 largest prime in regular forms
8D
28 = 256 hypercube vertices · the right-angled frame
2·8 = 16 axis-skeleton composite
7 largest prime in regular forms
9D
29 = 512 hypercube vertices · the right-angled frame
2·9 = 18 axis-skeleton composite
7 largest prime in regular forms
10D
210 = 1024 hypercube vertices · the right-angled frame
2·10 = 20 axis-skeleton composite
1111 enters largest prime in regular forms
11D
211 = 2048 hypercube vertices · the right-angled frame
2·11 = 22 axis-skeleton prime dimension
11 largest prime in regular forms
12D
212 = 4096 hypercube vertices · the right-angled frame
2·12 = 24 axis-skeleton composite
1313 enters largest prime in regular forms
13D
213 = 8192 hypercube vertices · the right-angled frame
2·13 = 26 axis-skeleton prime dimension
13 largest prime in regular forms

The Frame Built of the Primes

Two is the prime of perpendicularity. A dimension is an axis with two senses — forward and back — so adding a dimension doubles the right-angled frame, and the hypercube’s 2d vertices are nothing but the record of d perpendicular doublings of the point. The right angle has exactly one prime, and it is two.

The orthogonal skeleton carries the dimension-number itself (2·d), so each dimension is prime or composite in its own structure — the 2nd, 3rd, 5th, 7th, 11th, 13th being prime dimensions. And the regular forms admit the primes by threshold: only 2, 3, 5 through the fifth dimension, with seven first entering in the sixth, eleven in the tenth, thirteen in the twelfth. The primes do not cap dimension — they unfold it.

Read with the Corpus: perpendicularity, the relation between the unlimited and genuine otherness, has an arithmetic, and it is the doubling prime. Three-dimensional space — built of 2, 3, 5 and closed below 7 — is the last whose every regular form is the work of the Trinity’s primes and the microcosm’s. And seven, the completion the embodied cosmos cannot hold, is the first prime of the dimension beyond the solids — the Sabbath number that opens the sixth.

↓ Read the full paper (PDF)