It's easy to freehand a Voronoi diagram if you can hold the Delaunay triangulation in your head.

Solving a Voronoi using origami requires only one type of fold, and skips the Delaunay triangulation.

Humiaki Huzita's second origami axiom

Also using origami another delightful symmetry emerges:

Simultaneously creasing the three lines of a Voronoi intersection draws the three points to meet each other in space. Collapsing the Voronoi intersection flat will put a crease through each of the 3 points.

A. Add a line that runs parallel to a Voronoi wall segment but inset toward the point by a certain amount to your choosing.

B. Establish the endpoints at both the intersections to each edge crease.


Between neighboring cells, lines are copies of themselves reflected across the shared cell wall.

mountain valley crease pattern

It has this property: 

Each new boundary is same ratio distance between the point and the edge, along its perpendicular.

It also has this property:

Each smaller cell is an affine scale of its parent Voronoi cell with its homothetic center at the cell's point.

There are numerous ways to make the 3D back face lie flat. Squashing a triangle into an intersection gives equal president in Z-space to the 3 panels from the front perspective.

Triangles will most likely overlap too, it gets more complicated, but you can remedy them in a number of different ways.

Upper left: 4-way squash
Just to its right: 4-way fold over. no squash.
Center: squash, then fold over.
Bottom center: 2 triangles overlapping. squash together then fold to the side.


one beat, indivisible
whole above parts
order by number of beats
most even to least even, prioritize large first 
2^(N-1), where N is number of possible beats
six/eight time, or a waltz of eighth notes
as the scale increases, patterns and self-similarity become more apparent

a coloring book for J.S. Bach:
J.S. Bach, Violin Concerto in Am (BWV1041), violin solo
J.S. Bach, Brandenburg Concerto 4, violin solo
J.S. Bach, Invention 4 in Dm

shade indicates frequency (dark : high)
Erik Satie, Gymnopédie 1
Philip Glass, Metamorphosis 1

- subdivisions prefer to be center justified. this is a system aware of its boundaries, unlike binary counting, where the subdividing would be left or right justified.
- weighting ranks long notes before short notes, which reflects a study in which classical composers exhibit the same tendency.

* In this instance of Glass' piece, eight note oscillations are grouped as quarters, if this rule is embedded, no data is lost.


Fig. 1

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Matthew Flickinger