Flexagons and the Math Behind Twisted Paper

Below I’ve collected my notes taken during the FutureLearn 3-week online course “Flexagons and the Math Behind Twisted Paper” presented by Dr. Yossi Elran of Davidson Institute of Science Education, Weizmann Institute of Science. This was a fascinating introduction to Flexagons and symbols to describe them, Möbius band, Catalan numbers and more.

These are pretty rough notes at the moment, probably not very meaningful to those who haven’t taken this course. But stay tuned - I plan to flesh these notes out to an article with images and diagrams later.

Week 1

  • Flexagons were invented by Arthur Stone in 1939.

  • A hexagon shaped paper toy, when manipulated by a series of flexes, revealed unexpected properties.

  • He shared his discovery with his friends at Princeton: Bryant Tuckermann, Richard Feynman and John Tukey to form a flexagon committee to study the growing family of flexagons.

  • Flexagon diagrams used by the Flexagon committee were Feynman’s inspiration for his particle physics diagrams.

  • Tri-hexaflexagon: two sides (faces), a third face can be exposed with a pinch flex.

  • Thickness of flexagon alternates between stacks of 1 or 2 leaves. Stacks of leaves are called pats.

  • Tri-hexaflexagon has 6 pats, alternating between 1 or 2 leaves.

  • Naming system: Arthur Stone & friends used Greek prefixes to the word flexagon. Tri-hexa-flexagon has 3 faces (tri) and each face is 6-sided (hexagonal). Later the second prefix was used to denote the number of shapes (e.g. trianges) that make up each face.

  • Peter Hilton and Jean Pedersen replaced the greek prefixes with numbers in their book A Mathematical Tapestry. A tri-hexaflexagon is a 3-6-flexagon, a tetra-octaflexagon is a 4-8-flexagon.

  • Non-cyclic tetra-tetra-flexagon: two faces visible after assembly, book-flex to reveal the third, repeat to reveal fourth face.

  • Non-cyclic hexa-tetra-flexagon: two faces visible after assembly, book-flex can be applied vertically and horizontally to expose four more faces.

  • Every flexagon has a number of faces also referred to as states.

  • Every face divided into identical geometrical shapes.

  • Each of the visible leaves on the face is stacked as folded pats.

  • A flex changes state of flexagon.

  • Bryant Tuckerman designed a graph that could explain what happens when you flex a flexagon. These diagrams became known as Tuckerman diagrams.

  • States of the flexagons are at the vertices of the diagrams. The colour of the vertices is the color of the face-up state, the shadow color is the face-down state.

  • Arrows betwee two states represent the flexes. A label can be used to indicate type of flex. No label usually means pinch-flex.

  • Series of flexes that traverses the Tuckerman diagram completely by using the minimum number of flexes, is called the Tuckerman traverse.

Week 2

Week 3

  • A strip of paper has 2 sides. Attaching its ends together produces a band. The band also has 2 sides (inside, outside). Cutting along its center creates 2 new bands, each half the width of the original.

  • Making a half-twist before joining the ends creates a 1-sided Möbius band.

  • 1-sided bands can be obtained by making an odd number of half-twists.

  • 2-sided bands can be obtained by making an even number of half-twists.

  • Number of half-twists is the order of the band. 0-band (regular non-half-twisted), 1-band (Möbius), 2-band, 3-band, etc.

  • When you cut an n-band:

    • if n is odd: knotted band with total of 2n+2 half-twists
    • if n is even: 2 bands linked together n/2 times, each with n half-twists
  • 2-band cut through the middle = two 2-bands linked together once, with 2 half-twists

  • 3-band cut through the middle = knotted band with 8 half-twists

  • Boundary of 2-band is the Hopf link

  • Boundary of 3-band is the trefoil knot

  • Boundary of 4-band is Solomon’s knot

  • Recommended readings: website and Cliff Pickover’s book The Mōbius Strip

  • Cyclic flexagons are Möbius strips.

  • Tri-hexa-flexagon is a 3-band (straight strip of paper, half-twisted three times).

  • A Möbius band that has the same topological structure as a flexagon is its Möbius band twin.

  • Number of half-twists in the Möbius band twin of a straight strip cyclic flexagon with N faces and n triangles per face is 3N-n. Tri-hexa-flexagon with N=3 faces, n=6 triangles = 3*3-6 = 3-band twin.

  • Flexagons whose Möbius band twin has an even number of half-twists should have in theory two independent Tuckerman diagrams. Example hexa-hexa-flexagon is equivalent to 12-band. Performing V-flex iteratively transfers us from regular Tuckerman diagram to the diagram of the inverted strip while exchanging dominant and hidden faces.

  • 3-band cut in half gives an 8-band with a trefoil knot. But the trefoil knot can also be created directly by tying the strip into a knot before joining the end to a 1-sided band. Problem: 3-band alone is not a unique description.

  • Revised notation: paper abnd is prefixed with two symbols: 1) the number of knots crossings and sequential number according to Rolfsen’s knot table (01, 31, 41, 51, 52, etc), 2) total number of half-twists in the band. Examples:

    • Regular loop: 01-0-band
    • Möbius strip: 01-1-band
    • Tri-hexa-flexagon: 01-3-band
    • The 8-band with trefoil knot in it: 31-8-band
  • This notation works well for describing bands with twists and knots, but it’s not complete. It doesn’t account for multiple knots in a band or deal with pat structures in flexagons.

  • The Schläfli symbol for regular polygons is the number of vertices (or edges) written in curly brackets. {3} for (equilateral) triangle, {4} for a square, {5} for a regular pentagon, {6} for a regular hexagon, etc.

  • The Schläfli symbol for a star polygon is a fraction inside curly brackets. Numerator is number of vertices, denominator is the number of vertices skipped when drawing each edge of the star plus 1. {52} for example is a pentagram. {93} is a nonagram (9-pointed star).

  • Les Pook’s flexagon symbols use the Schläfli symbol. It has 3 components:

    • Number of sectors of the flexagon. Tri-hexa-flexagin is made of 3 identical sectors of 2 equilateral triangles joined at a common edge. Each sector has 2 pats, 1 with 1 leaf, the other with 2 leaves. Difference in number of leaves is the reason why there are 3 sectors, not 6. Hexa-hexa-fleagon has 3 sectors as well.
    • Schläfli symbol of the polygon of the flexagon, the basic shape. The basic shape that makes tri-hexa- and hexa-hexa-flexagon is a triangle {3}
    • Flexagon net symbol, Schläfli symbol of a polygon that describes dynamics of flexagon.
  • For the tri-hexa-flexagon the flexagon symbol is: 3<3,3> The magic square flexagon symbol is: 2<4,4>

  • Sector symbols: Notation starts off the same way as flexagon symbol, but ommitting the first component and adding 2 additional numbers for number of leaves in each pat. For tri-hexa-flexagon <3,3,2,1>.

  • Sector symbol for magic square flexagon: <4,4,3,1>

  • Flexagon symbol for hexa-tetra-flexagon: 2<4,4>

  • Catalan Numbers: 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862… For a grid with dimensions n * n, the nth Catalan number gives the number of paths from one corner to another across the grid. For a grid of 4 * 4, the number of paths is the 4th Catalan number, 14.

  • Catalan numbers can also be used to answer the question how many ways there are to divide a polygon with n+2 sides into non-crossing triangles. For a hexagon (4+2), it’s the 4th Catalan number, 14. Further reading: brilliant.org

  • Catalan numbers show up in formulas for calculating the number of different kinds of flexagons, and in the number of ways pats can be split into sub-pats, e.g. a 4-leaf pat can be split in 5 different ways. (n+1) leaf pat can be split in the nth Catalan number of ways.

Some further thoughts and questions

I was curious and searched for examples of knots in nature. I found so many examples, from quantum knots, knots in chaotic waves, in fluid dynamics… I also found this paper on molecular knots in biology and chemistry. It includes some nice examples of the knot notation I learned about in the course as well. I can only understand a little of the tip of an iceberg here, but it makes me wonder if the spark of life was thanks to a knot and the properties of a half-twisted band. It definitely made me appreciate the importance of understanding them if one wants to understand life, existence and nature in its entirety.

With all the tools and symbols we have available to describe Flexagons, sections and leaf pats now, I’m wondering… do they also find practical applications in areas beyond recreational math, e.g. in computational geometry or computer graphics, or in engineering, say, when some material is folded in certain ways to achieve certain strength, or even areas that don’t directly relate to geometry?