Algebraic Biology · gallery

The vivoids,
and their time

Four renderings of the same Weyl orbit of E₈ — 2160 centers on a single sphere. The form comes from the powers; the color, in each one, is the time. Two of them count it in space, two in the algebra: it is the only difference, and it is the whole difference.

2160centers

240roots of E₈

ℓ = 0 → 78weak order

12files

ℓ = 0 · the dominant62 centers at ℓ = 39ℓ_max = 78

01 — Point cloud

ALFA

zspatial time

The centers of the orbit, not yet clothed in a surface. Emergence follows the axis coordinate: the eye's rendering, not the group's.

ALFA — still — StillThe main views, side by side.

ALFA — growth — GrowthThe body unfolding.

ALFA — rotation — RotationThe mature body, turning.

02 — Surface

BETA

zspatial time

The same body, closed by a lit membrane. Time is still spatial: it grows from bottom to top, not according to the algebra.

BETA — still — StillThe main views, side by side.

BETA — growth — GrowthThe body unfolding.

BETA — rotation — RotationThe mature body, turning.

03 — Surface

GAMMA

ℓalgebraic time

The faithful version. Color and growth follow the length ℓ in the weak order: the body emerges from the dominant, one reflection at a time.

GAMMA — still — StillThe main views, side by side.

GAMMA — growth — GrowthThe body unfolding.

GAMMA — rotation — RotationThe mature body, turning.

04 — Rich form

DELTA

ℓalgebraic time

A bulbous head, a trunk, a tapering tail, bilateral symmetry, a hint of segmentation. The morphology closest to an expected organism.

DELTA — still — StillThe main views, side by side.

DELTA — growth — GrowthThe body unfolding.

DELTA — rotation — RotationThe mature body, turning.

The color is the time. In Alfa and Beta it follows the spatial coordinate of the axis: the body grows from bottom to top, but that order does not come from the theory. In Gamma and Delta it follows the length ℓ in the weak order — the order of deployment the group prescribes. Same form, two times: only the algebra is faithful to the model.

The engine behind the forms

The forms on this page are generated by the Weyl Machine: a device that translates in both directions between a compact algebraic specification and a structured form. Its inverse mode is implemented as a variational quantum algorithm in Q#, targeting quantum hardware.

Provisional patent application filed with the USPTO — No. 64/111,888, filed July 15, 2026. The patent covers the technical procedure only, and does not depend on the validity of the biological model.