Rhea

Advanced mode

Conway operators

Nineteen polyhedral operators that chain together, turning simple solids into forms with no common names. No maintained Blender tool does this properly, which is why it is here — and there is a picker, so you never have to learn the notation to use it.

What this actually is

Conway notation is a way of writing down transformations of polyhedra. Each operator is a letter, each letter is a rule for rebuilding a mesh from its own topology, and the rules compose. A truncated icosahedron — tI — is a football. dkD is something with no name at all.

The reason to care in a design context is that these produce topologically clean, evenly distributed, structurally sensible meshes that would be extremely tedious to model by hand. A geodesic sphere is a Conway chain. So is most of the panelisation you see on modern facades.

You do not need to know the notation

The section has a visual picker: every operator shown as a thumbnail with its name and what it does. Click one and it is added to the chain. Click Undo step to remove the last one, or Clear to start over.

  1. 01

    Pick a seed

    Tetrahedron, cube, octahedron, dodecahedron, icosahedron, or Use selected object to run operators on your own mesh.

  2. 02

    Click operators

    Each click adds a letter and rebuilds. Watch what happens; that is faster than reading about it.

  3. 03

    Adjust the per-operator parameters

    When the chain contains an operator with a shape parameter — kis apex height, truncate ratio, chamfer inset, bevel width, loft depth — a slider appears for it, indexed by its position, so the same operator used twice gets two independent controls.

Reading the notation

Operators apply right to left, and an uppercase letter at the end is the seed. dkD means: build the Dodecahedron, apply kis, then apply dual. It reads backwards because it is written the way function composition is written in mathematics.

The nineteen operators

d — dual
Every face becomes a vertex and every vertex becomes a face. Applied twice it returns the original, which is the standard sanity check.
a — ambo
Vertices at every edge midpoint. Turns a cube into a cuboctahedron. The most useful single operator for opening a form up.
k — kis
Raises a pyramid on every face. The apex height is adjustable, and at zero it does nothing visible — worth knowing, because a kis that appears to have failed is usually a kis with its height left low.
t — truncate
Cuts every vertex off. A truncated icosahedron is a football. The ratio controls how deep the cut goes.
e — expand
Pushes every face outward and fills the gaps with squares. Produces the rhombi-forms.
o — ortho
Subdivides every face into quads. The quad-producing operator, and the one to reach for before anything that needs a quad mesh.
j — join
Rhombic faces spanning each original edge. Turns a cube into a rhombic dodecahedron.
s — snub
Twists every face and fills with triangles, producing a chiral result — it has a handedness, unlike almost everything else here.
g — gyro
Pentagons arranged with a twist. Also chiral, and the source of the most distinctive Conway pattern.
m — meta
Triangles from face centres to edge midpoints. Fine subdivision with a radial character.
b — bevel
Truncate and ambo combined. Produces hexagons and squares, the classic bevelled solid.
n — needle
The dual of kis: raises spires from a different starting arrangement. Sharp and spiky.
z — zip
The dual of truncate. Where truncate cuts corners, zip cuts across faces.
c — chamfer
Inserts hexagons along every edge without moving the original faces. The best operator for adding density while keeping the silhouette.
p — propeller
Rotated quads around each face, producing a pinwheel. Chiral, and unmistakable once you have seen it.
w — whirl
Hexagons spiralling around each face. The most decorative operator in the set.
l — loft
Raises a smaller copy of each face above it, joined by a band. Depth is adjustable. Produces panelised, layered results.
q — quinto
Pentagons from edge midpoints. Gives an organic, cellular character that gyro shares but from a different construction.
r — reflect
Mirrors the form, converting a chiral result to its opposite hand. Only meaningful after snub, gyro, propeller or whirl.

Canonicalize

Raw Conway output is topologically correct and often visually lumpy — faces are not planar, the form is not centred, and it does not look like the regular solid you were expecting.

Canonicalize fixes that with iterative spherical relaxation: planarise faces, recentre, normalise radius, repeat. It is off by default because it costs time, but it is the difference between “a mesh with the right topology” and “the shape people picture when they hear geodesic sphere”.

Iterationsup to 500
How many relaxation passes. Fifty is usually plenty; several hundred is for forms that start badly distorted. Diminishing returns set in fast.
Shade smoothangle
Smooths only where faces meet below this angle, so hard polyhedral edges stay hard. Without it the whole result renders as a soft blob and every operator looks identical.

If every chain looks the same

Three things cause this, and it is usually all three at once: shading is fully smooth so edges are invisible, canonicalize is off so the forms are lumpy, and kis height is left near zero so the operator does almost nothing. Turn on angle-based smoothing first — it fixes the most.

Face counts grow fast

  C     cube                       6 faces
  tC    truncated cube            14
  ktC   kis of that               56
  aktC  ambo of that             112
  gaktC gyro of that             672

Five operators takes a cube past six hundred faces; on a dodecahedron the same chain lands in the tens of thousands. Rhea precomputes the projected face count before running and refuses above the vertex budget with a message naming the number, rather than freezing.

Input requirements

  • Manifold input. Conway operators are defined on closed surfaces. Non-manifold input is detected and refused by name rather than producing garbage.
  • No boundaries. An open mesh has faces with edges that belong to only one face, and most operators have nothing sensible to do with those.
  • Consistent winding. Recalculate normals on an imported mesh before running a chain on it.

The twenty-six presets

Every named solid people actually want, plus the ones worth discovering: soccer ball, geodesic sphere, cuboctahedron, icosidodecahedron, the four truncated Platonics, both rhombic solids, rhombicuboctahedron, rhombicosidodecahedron, snub cube, snub dodecahedron, pentakis shell, star shell, bevelled shell, chamfered cube, needle spire, lofted cage, meta web, gyro pattern, propeller twist, whirl spiral, quinto shell, rhombic cells, rhombic triacontahedron.

Applying a preset sets the notation, the seed, the per-operator parameters and the canonicalize settings. It does not run — press the button yourself.

Where this fits

A Conway result is an excellent base for everything else in Rhea, because it is exactly the kind of mesh the other modules want: even faces, no poles, clean topology, no boundaries.

  • ortho anything to get quads, then run Patch-mode tessellation over it.
  • Build a geodesic sphere, then apply the Uniform honeycomb effect for a panelised dome.
  • Run form finding on a Conway result to relax it into a shape that carries load.
  • kis a truncated icosahedron and tessellate spikes onto the result for something genuinely unpleasant to hold.