Superformula · Herbarium Syntheticum

Herbarium syntheticum · plate I · pressed 2026

One equation,
a thousand silhouettes

The Gielis superformula, mounted and tagged like flora.

The specimen sheet

12 pressed forms · 1 living cell · a = b = 1 · tap a specimen to summon it

SF-00 · VIVUM

touring parameter space

m5 n₁2.00 n₂7.00 n₃7.00

Specimen vivumthe living cell — it never stops growing

SF-01

Circulus omniumthe null case — every tour passes through it

m 4 · n₁ 2 · n₂ 2 · n₃ 2

SF-02

Trigona lenisa triangle grown, not ruled

m 3 · n₁ 4.5 · n₂ 10 · n₃ 10

SF-03

Quadrata firmathe square, coaxed from a circle

m 4 · n₁ 12 · n₂ 15 · n₃ 15

SF-04

Pentagona verafive sides, no straightedge

m 5 · n₁ 7 · n₂ 7 · n₃ 7

SF-05

Stella quinqueradiatastarfish — drop n₁ and the arms reach

m 5 · n₁ 2 · n₂ 7 · n₃ 7

SF-06

Hexagona cellaethe honeycomb wall

m 6 · n₁ 9 · n₂ 14 · n₃ 14

SF-07

Astroidea acutathe astroid — n below one pulls spikes

m 4 · n₁ 0.5 · n₂ 0.5 · n₃ 0.5

SF-08

Corolla octopetalaeight petals from an unequal n-pair

m 8 · n₁ 0.5 · n₂ 0.5 · n₃ 8

SF-09

Diatoma radiansa diatom's scalloped valve

m 16 · n₁ 0.5 · n₂ 0.5 · n₃ 16

SF-10

Rota dentatanineteen teeth — botany becomes machinery

m 19 · n₁ 9 · n₂ 14 · n₃ 11

SF-11

Semen unicumm = 1 — the lone seed leans

m 1 · n₁ 0.8 · n₂ 0.8 · n₃ 0.8

SF-12

Echinus duodeciman urchin's twelve-spined test

m 12 · n₁ 0.6 · n₂ 0.9 · n₃ 0.9

In 2003 the Belgian botanist Johan Gielis noticed that one small edit to the circle's equation lets it grow corners, petals, spines and teeth. Feed it five and a starfish opens; feed it nineteen and it cuts a gear.

Every outline on this sheet is that single formula wearing different numbers. Nothing is drawn by hand — each specimen is pressed straight from the mathematics, and its tag records the exact parameters that grew it.

Reading the tags

msymmetry
How many times the form repeats around the circle. m = 5 opens a starfish; m = 19 cuts a gear.
n₁tension
The outer exponent. Push it below 1 and the outline is sucked inward into spikes; raise it and the form inflates toward its bounding polygon.
n₂ n₃the corner-makers
They govern the cosine and sine lobes separately. Keep them equal for even forms; split them and petals bloom where corners were.
a bthe two radii
Held at 1 across this plate, so every specimen sits in the same press. Stretch one and the whole flora leans.