Universe

A zero-dependency Typst package that draws the famous spiral of right triangles. Each triangle adds a leg of length size at a right angle to the previous hypotenuse, so the n-th triangle has legs √n·size and size, and hypotenuse √(n+1)·size.

#import "@preview/pythagorean-spiral:0.1.0": *

#pythagorean-spiral(steps: 17, size: 1cm)

Everything is drawn with plain Typst primitives (polygon, line, curve, circle, place) — no plugin, no CeTZ dependency, fully measurable, works anywhere a box does.

Parameters

Parameter Default Description
steps 12 Number of triangles (1–5000)
size 1cm Length of the starting leg
fill none Fill of the triangles: none, a color, a gradient (sampled along the spiral) or an array of colors (interpolated along the spiral)
stroke black Outline: none, a color, or a full stroke
stroke-width 1pt Outline thickness when stroke is a color
direction "ccw" "ccw" (counter-clockwise) or "cw"
start-angle 0deg Rotation of the whole spiral
mode "triangles" "triangles", "spiral" (outer staircase only) or "rays" (segments from the centre only)
show-hypotenuses true Draw the inner rays O → Pₙ
right-angle-marks false Draw small square marks at every right angle
right-angle-size 3mm Side of the right-angle marks
center-dot true Dot at the centre O
center-dot-radius 1.2mm Dot radius
center-dot-fill black Dot colour
labels none "vertices" (P₀…Pₙ), "indices" (numbers the triangles) or "lengths" (leg/hypotenuse lengths)
label-size 8pt Label font size
label-offset 4mm Distance of vertex labels from the vertex
length-labels none "values" (decimal of a·√n, e.g. “1.414 cm”), "formulas" (exact √n in units of a, simplified: √1→1, √4→2), or a function (k, len) => content — writes the exact length a·√n of each ray; labels are rotated parallel to their ray and centred on it
label-new-legs false Also label the new perpendicular legs of every triangle (each has the exact length a of the starting leg); the labels are placed on the OUTSIDE of the spiral so they never cover the drawing
new-leg-offset 2.5mm Perpendicular distance of the new-leg labels from their segment (outward)
length-size 7pt Font size of the length labels
length-pos 0.72 Position of the ray labels ALONG their ray: 0 = at the centre O, 1 = at the outer endpoint Pₙ (near the outer end the rays are further apart, so labels do not overlap)
length-offset 0 Perpendicular distance of the ray labels from their ray (0 = labels sit on the segment)
length-label-background white Background colour behind the length labels (white by default so labels stay readable over the strokes, even without triangle fills; pass none to remove it)
padding 7mm Space around the drawing
background none Canvas background colour

See manual.pdf

Examples

// classic classroom spiral
#pythagorean-spiral(steps: 17, size: 1cm)

// gradient fill, clockwise, thicker stroke
#pythagorean-spiral(
  steps: 30,
  size: 1.2cm,
  fill: gradient.linear(rgb("#ff6b6b"), rgb("#4ecdc4")),
  stroke: 1.5pt + rgb("#2d3436"),
  direction: "cw",
)

// hand-coloured triangles + right-angle marks + indices
#pythagorean-spiral(
  steps: 10,
  size: 1cm,
  fill: (yellow, orange, red),
  right-angle-marks: true,
  labels: "indices",
)

// length labels: the exact length a·√n of every ray
// (n = 1 is the starting leg, length a = a·√1)
#pythagorean-spiral(
  steps: 8, size: 1.2cm,
  fill: blue.transparentize(70%),
  length-labels: "values",       // decimal: 1 cm, 1.414 cm, 1.732 cm, ...
  length-label-background: white,
)
#pythagorean-spiral(
  steps: 8, size: 1.2cm,
  length-labels: "formulas",     // exact: √1, √2, √3, ...
  label-new-legs: true,          // the new legs too: each has length a -> "1"
)

// poster with vertex labels
#pythagorean-spiral(
  steps: 8,
  size: 1.4cm,
  labels: "vertices",
  label-size: 9pt,
  fill: blue.transparentize(70%),
)

Geometry helpers

#spiral-points(steps)        // vertices P_0 … P_N (math coords, units of size)
#spiral-angle(steps)         // accumulated angle (angle)

With 17 triangles the accumulated angle already exceeds a full turn: #spiral-angle(17) ≈ 364.78° — the spiral famously overshoots 360°.

License

MIT. This is an independent package, not affiliated with any LaTeX project.

FERGOUS Abdelhak