Geometry · nature · pattern

Change it.
Keep it.

Symmetry is the study of transformations that leave something essential unchanged.

The governing idea

A symmetry always asks two questions.

01What transformation are we allowed to perform?

02What must remain invariant afterward?

Transformation 01

Reflection symmetry

MₙX = X

A line or plane divides a form into matching mirror images.

What stays invariant: Outline, spacing, angles, and visual weight on opposite sides of the mirror axis.
Transformation: Mₙ(x) = x − 2[(x − c) · n]n

Precision noteNatural forms are usually approximate; designed forms can be mathematically exact.

Artifact 01

Butterfly

Paired wings mirror a central body axis.

Artifact 02

Human face

A familiar example of approximate bilateral form.

Artifact 03

Facade

Doors, windows, and mass balance around a center.

Artifact 04

Leaf

The midrib acts as a natural reflection line.

Artifact 05

Vessel

A profile mirrored around its vertical centerline.

Transformation 02

Rotational symmetry

RθX = X

Turning a form around a fixed center by a specific angle returns the same appearance.

What stays invariant: The placement and orientation of repeated parts around the center.
Transformation: n-fold symmetry: θ = 360° / n

Artifact 01

Square

Order 4: unchanged at every quarter-turn.

Artifact 02

Snowflake

Six branches establish order-6 symmetry.

Artifact 03

Propeller

Three blades repeat at 120° intervals.

Artifact 04

Mandala

Concentric motifs repeat around one center.

Artifact 05

Rosette

Eight petals create an order-8 ornament.

Transformation 03

Translational symmetry

TₐX = X

Sliding a motif by a fixed distance reproduces the same visual pattern.

What stays invariant: The motif, its orientation, and spacing within each repeat unit.
Transformation: Tₐ: x ↦ x + a

Artifact 01

Wallpaper

A motif repeats in two independent directions.

Artifact 02

Brick wall

Regular units shift across staggered rows.

3D · drag

Artifact 03

3D lattice

Points repeat through three spatial directions.

Artifact 04

Woven textile

Over-under strands generate a repeating grid.

Artifact 05

Border pattern

One motif travels along a single direction.

Transformation 04

Glide-reflection symmetry

Gₐ = TₐM

Mirror a motif, then slide it parallel to the mirror axis to recover the pattern.

What stays invariant: The alternating sequence after both reflection and translation are applied.
Transformation: Gₐ = Tₐ ∘ M

Artifact 01

Footprints

Left becomes right and advances one step.

Artifact 02

Triangle frieze

Alternating forms flip across a traveling axis.

3D · drag

Artifact 03

3D glide plane

A plane reflection combines with an in-plane shift.

Artifact 04

Swimming fish

Alternating fish reverse and move along the band.

Artifact 05

Flying birds

A reflected silhouette continues the visual rhythm.

Transformation 05

Point symmetry

r ↦ −r

Every feature has a matching feature directly opposite a central point.

What stays invariant: Distance from the center and the full arrangement after a half-turn.
Transformation: x ↦ 2c − x · equivalent to 180° in 2D

Artifact 01

Cubic curve

The curve maps through the origin onto itself.

Artifact 02

Parallelogram

Opposite corners pair through the center.

Artifact 03

Hexagonal ring

Opposite vertices match across the center.

Artifact 04

Playing card

The top and bottom designs exchange on a half-turn.

Artifact 05

Yin-yang

Two lobes exchange positions under 180° rotation.

Transformation 06

Helical / screw symmetry

Sθ,h

A turn around an axis combines with movement along that axis to preserve a helical form.

What stays invariant: The pitch, radius, and repeating relationship between rotation and rise.
Transformation: (r, φ, z) ↦ (r, φ + θ, z + h)

3D · drag

Artifact 01

Interactive helix

Drag to inspect constant rotation plus rise.

Artifact 02

Machine thread

Each complete turn advances by one pitch.

3D · drag

Artifact 03

Helical stair

Repeated steps climb around a central axis.

3D · drag

Artifact 04

Double helix

Two paired strands wind around one axis.

Artifact 05

Spiral shell

Rotation combines with growth along a coiled path.

Transformation 07

Scale / self-similarity

x ↦ λx

Zooming in or out reveals the same form, rule, or visual character at another size.

What stays invariant: Shape, proportions, branching rule, or statistical roughness.
Transformation: f(λx) = λᵟ f(x)

Precision noteFractals can be exact; leaves and coastlines show approximate self-similarity over a limited range.

Artifact 01

Sierpiński triangle

Three reduced copies recreate the whole.

Artifact 02

Koch curve

Every segment repeats the same four-part rule.

Artifact 03

Coastline

Roughness looks statistically similar across scales.

Artifact 04

Fern

Small fronds echo the outline of the full leaf.

Artifact 05

Nested polygons

A constant scale and turn generate each copy.

Transformation 08

Polyhedral symmetry

G ⊂ SO(3)

A three-dimensional solid admits several rotations that exchange identical faces, edges, and vertices.

What stays invariant: The solid's spatial arrangement and incidence of faces, edges, and vertices.
Transformation: T, O, and I rotation groups

3D · drag

Artifact 01

Tetrahedron

Four equivalent triangular faces and vertices.

3D · drag

Artifact 02

Cube

Six faces exchange through several rotation axes.

3D · drag

Artifact 03

Octahedron

The cube's dual shares the same rotation group.

3D · drag

Artifact 04

Icosahedron

Twenty faces produce rich fivefold and threefold axes.

Artifact 05

Kaleidoscope

Repeated mirror wedges suggest polyhedral order.