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A New Twist on Flexagons?

October 22, 2014

For Martin Gardner’s 100th birthday

Martin Gardner
Global Science source

Martin Gardner introduced many including myself to the joys of Discrete Mathematics. His glorious monthly column “Mathematical Games” for Scientific American included some continuous mathematics too, of course; one could say it was on “Concrete Mathematics.” However, I conjecture—based on a quick flip of the several books I own of his columns—that the symbols {\epsilon,\delta} in a calculus context never appeared in them.

Yesterday was the 100{{}^{th}} anniversary of Gardner’s birth. Dick and I wish to join the many others marking this centennial and thanking him for all he did to make math fun for so many.

His feature kicked off in 1956 with the famous column on hexaflexagons, which I will talk about in a moment. Gardner related in his autobiography, which was assembled three years after his death in 2010, how important this column was as his “break.” However, the column that made the most lasting impression on me began with the words:

The calculus of finite differences, a branch of mathematics that is not too well known but is at times highly useful, occupies a halfway house on the road from algebra to calculus.

This discrete “calculus” enables one to calculate a formula for any polynomial sequence {p(n)} given enough values for {n = 0,1,2,...} It also led to my favorite “visual proof” that {0^0 = 1}: For any integer {k > 0}, if you write out the powers {k^n} going across and take differences of adjacent values repeatedly to make an infinite equilateral triangle pointing down, the left side has the powers of {(k-1)}. Iterating this gives you the powers of {0}, but the entry for {k^0} as {k} counts down to {0} steadfastly remains {1}.

Tributes and Contributions

Tributes have been gathered all during this centennial year. Scientific American observed yesterday by posting a review of ten of Gardner’s most appreciated columns. Bill Gasarch’s post yesterday links to some of his and Lance Fortnow’s previous items on Gardner, and further to a site where anyone can contribute a testimonial.

Frederic Friedel, who co-founded the chess-database company ChessBase three decades ago, knew Gardner personally from 1979 as a fellow original member of the Committee for Scientific Investigation of Claims of the Paranormal (CSICOP, now CSI). The committee remains housed in my town of Amherst near Buffalo, now at the Center for Inquiry (CFI Western NY) which is across Sweet Home Road from my university campus. Friedel has described to me cold days in Buffalo and round tables with Carl Sagan and other luminaries. All this was before my own arrival in 1989.

Friedel was also among the column readers with whom Gardner interacted from the beginning in the 1950s. His awesome tribute yesterday includes appreciation of Gardner’s book Fads and Fallacies in the Name of Science, which also made a strong impression on me, and other links. Dick recalls the great chapter of that book that starts with Gardner saying that this next crazy claim cannot be disproved. It was that the universe was created recently with a full fossil record that makes it look much older. Indeed, it could be a so-called “Boltzmann Brain”—and a point made in this NY Times article is that it’s crazy that this is not crazy.

I never had any contact with Gardner, despite making a few visits to CFI; it ranks among numerous lost opportunities. I could mention many other influences from his columns, and looking through his book Mathematical Circus just now reminded me that his chapter on “Mascheroni Constructions” was my first knowledge of what I called the “class {\mathsf{NC}}” in my “STOC 1500” post with Dick. I had a similar experience to what Douglas Hofstadter told in his own tribute in May 2010: I started off interested in Math+Physics, intending to take the latter as far as quantum mechanics and particles at Princeton. But I found advanced mechanics and electrodynamics tough going, and am ever grateful for being allowed to parachute out of the latter at midterm into Steve Maurer’s Discrete Mathematics course, in which I knew I’d found my métier. As I could have realized from my love of Gardner all along.

The Twist

I’ve wanted to make a post on my hexaflexagon twist when I had time to create nice pictures, but now I will have to make do by referring to the fine illustrations in Gardner’s original column, which is freely available from the M.A.A. here. It requires making the standard “hexa-hexa” as shown in Gardner’s Figure 2. For best effect, in addition to numbering the faces 1–6 as shown there (and using a solid color for each face), label the six components of each face A–F in the left-to-right order given there.

The “Twist” is always applicable from one of the three inner faces (1, 2, or 3); finding when it applies from one of the outer faces and from the configurations that follow is more of a challenge. Instead of flexing as shown in Figure 3, follow these directions:

  • Put your right thumb and forefinger on the two rightmost triangles (which will be ‘2A’ and ‘2B’ in the bottom part D of Figure 2 after the final flap with ‘1A’ is folded to complete the flexagon), and put your left thumb and forefinger over the opposite two (‘2E’ and ‘2D’).

  • Fold the flexagon in half away from you, keeping the horizontal diameter toward you.

  • Then tease the bottom-left triangle and the bottom triangle (‘2D’ and ‘2C’) toward you to open a bowl of four triangles, with your right thumb and forefinger now pinched together to make a flap that stands over the bowl.

  • Fold the flap over into the bowl (‘2B’ onto ‘3B’), and then fold the bowl flat with the top point meeting the bottom point. You now have a lozenge of two triangles (‘2E’ and ‘2F’ uppermost).

  • Flip the lozenge over so you can grab at the opposite point from the two that met.

  • You should be able to open from that point into another bowl of four triangles.

  • Then you can lift the thickest flap at right (with ‘1C’) out of the bowl. (If a twist move fails, this is where it fails, in which case backtrack and try again elsewhere.)

  • Then close the bowl inversely to how you opened the other bowl, so that you again have a flexagon folded in half away from you, and grab at the opposite end to open it.

What you will get is a flexagon with the colors on its faces jumbled up—if you’ve used the lettering, you will have ‘1C’, ‘5B’, and ‘2F’-‘2E’-‘2D’-‘2C’ clockwise from upper right. You will still be able to flex it the standard way, but only exposing one other face—that is, you will have something isomorphic to a tri-hexaflexagon.

Now the real fun is that you can iterate this process. For one thing, you can invert it to restore your original hexa-hexaflexagon (teasing ‘2E’ and ‘2F’ forward and folding in ‘1C’). But you can also find other places from which to initiate another “Twist,” and these will lead to more tri-hexa configurations. One is to flip it over, rotate once counterclockwise so you fold backwards with ‘6B’ and ‘3C’ at right, tease forward ‘3E’-‘3D’, tuck ‘3C’ into the bowl atop ‘1D’, collapse and grab at the other end of ‘2A’-‘6A’, lift flap ‘2D’ out of the bowl, and unfold to see ‘2D’-‘4D’-‘6A’-‘2A’-‘3E’-‘3D’. Then you can flip over, rotate once more counterclockwise, and iterate—but there are other twists too.

Some will lump up thick wads of paper on three triangles of each face, so be ginger about it. Finally, after much exploration, you may come upon the “Dual Hexa.” This has six faces, in which the inner three alternate colors. It is, in fact, the configuration you would build if you first rotated the top part A of Gardner’s Figure 3 by 180 degrees. Then you may find a way to go from the primal to the dual and back by a long regular pattern of repeated twists.

As a high-school student in 1976, I attempted to map out the entire space of reachable configurations by hand, but made some bookkeeping errors and gave up. I wanted to write a computer program to simulate my twists, but did not make the time.

Open Problems

Can you do the “Twist”? The space of configurations you can explore is much larger than the “Tuckerman Traverse” of the standard hexa-hexa shown in Gardner’s Figure 4. Can you map it all out? Has anyone previously known about this?

[some format and word changes, updated to include letters of facets.]

9 Comments leave one →
  1. October 23, 2014 2:05 am

    I have two connections with Martin Gardner. The first one his interest in one of my conjecture on graceful trees that not only is every tree graceful, every tree has a graceful labelling in which the labels on each path alternately increase and decrease (see Knuth letter)

    Dancing  with Conjectures

    The strong graceful tree conjecture is still wide open.

    My second involvement is related his famous “counter-example” map for the four color conjecture. My solution to his map not only color it with four color but also decompose the regions into form of two spiral chains (double-spiral). This has been noticed by Chair of the Martin Gardner Centennial Committee Colm Mulcahy (Spelman College) and reserved to be presented in the new Gardner webpage. (see the links below):

    Strong four coloring of Martin Gardner's Map

    and the last slide before the references

  2. October 23, 2014 3:26 am

    My daughter discovered this by accident. We were playing with a hexahexaflexagon I’d made, and I was very surprised when she showed me that she’d got the colours mixed up. It took us quite a while to work out how to do it repeatably.

    I hoped she had made a new discovery, but of course flexagon enthusiasts know all about these flexes already. Here is a short video showing different modes of flexagation: I think the one you describe above is the “tuck flex”.

    I haven’t come across a map of the configurations. I’d love to see one.

    • October 23, 2014 8:16 am

      The “tuck flex” is not quite the same—it has another move and folds in a flap of two adjacent triangles rather than just one. But it is great to see all the other moves in the video…

  3. phomer permalink
    October 23, 2014 11:14 am

    A rather amusing discussion/video on Hexaflexagons comes from the rather energetic and always entertaing Vi Hart:


  4. December 8, 2014 9:36 am

    I can’t remember when I first started playing with Gödel codings of graph-theoretic structures, which arose in logical and computational settings, but I remember being egged on in that direction by Martin Gardner’s 1976 column on Catalan numbers, planted plane trees, polygon dissections, etc.

    Codings being injections from a combinatorial species \mathcal {S} to integers, either non-negative integers \mathbb{N} or positive integers \mathcal{N}. I was especially interested in codings that were also surjective, thereby revealing something about the target domain of arithmetic.

    The most interesting bijection I found was between positive integers \mathcal{N} and finite partial functions from \mathcal{N} to \mathcal{N}. All of this comes straight out of the primes factorizations. That type of bijection may remind some people of Dana Scott’s D_\infty. Corresponding to the positive integers there arose two species of graphical structures, which I dubbed riffs and rotes.

  5. Juh Raffe permalink
    November 1, 2017 10:44 pm

    In response to the open problem of mapping out the Twist flex, I believe there are 55 different states you can reach from the normal starting position of the symmetric hexahexaflexagon using just the twist flex. Below is the mapping between these states. The states are numbered 0 through 54. The letters describe the orientation: a through f are rotations (a is no rotation, b is 60 degrees cw, c is 120 degrees cw, etc.), and g through l are a flip followed by a rotation. The starting position/orientation is 0a.

    I drew a graph of the connections. There are six paths from state 0, where each path goes two hops and then splits. On each path, one split terminates while the other continues another 6 hops. The second to last hop of each of the six paths connects to the same position on another path, giving the overall pattern 3-way symmetry.

    The pinch flex has a nice pattern to the flex map as you double the hexaflexagon from 3 to 6 to 12 sides and beyond. I can’t find any such pattern for the twist flex. For the dodecahexaflexagon I could not find all of the states, but the 25,000 or so that I did find suggest that there is no simple pattern.

    0b –> 1a
    0d –> 2a
    0f –> 3a
    0g –> 4a
    0i –> 5a
    0k –> 6a
    1a –> 0b
    1l –> 7a
    2a –> 0d
    2l –> 8a
    3a –> 0f
    3l –> 9a
    4a –> 0g
    4l –> 10a
    5a –> 0i
    5l –> 11a
    6a –> 0k
    6l –> 12a
    7a –> 1l
    7k –> 13a
    7l –> 14a
    8a –> 2l
    8k –> 15a
    8l –> 16a
    9a –> 3l
    9k –> 17a
    9l –> 18a
    10a –> 4l
    10k –> 19a
    10l –> 20a
    11a –> 5l
    11k –> 21a
    11l –> 22a
    12a –> 6l
    12k –> 23a
    12l –> 24a
    13a –> 7k
    14a –> 7l
    14l –> 25a
    15a –> 8k
    16a –> 8l
    16l –> 26a
    17a –> 9k
    18a –> 9l
    18l –> 27a
    19a –> 10k
    20a –> 10l
    20l –> 28a
    21a –> 11k
    22a –> 11l
    22l –> 29a
    23a –> 12k
    24a –> 12l
    24l –> 30a
    25a –> 14l
    25l –> 31a
    26a –> 16l
    26l –> 32a
    27a –> 18l
    27l –> 33a
    28a –> 20l
    28l –> 34a
    29a –> 22l
    29l –> 35a
    30a –> 24l
    30l –> 36a
    31a –> 25l
    31l –> 37a
    32a –> 26l
    32l –> 38a
    33a –> 27l
    33l –> 39a
    34a –> 28l
    34l –> 40a
    35a –> 29l
    35l –> 41a
    36a –> 30l
    36l –> 42a
    37a –> 31l
    37l –> 43a
    38a –> 32l
    38l –> 44a
    39a –> 33l
    39l –> 45a
    40a –> 34l
    40l –> 46a
    41a –> 35l
    41l –> 47a
    42a –> 36l
    42l –> 48a
    43a –> 37l
    43d –> 49a
    43l –> 48l
    44a –> 38l
    44d –> 50a
    44l –> 47l
    45a –> 39l
    45d –> 51a
    45l –> 46l
    46a –> 40l
    46d –> 52a
    46l –> 45l
    47a –> 41l
    47d –> 53a
    47l –> 44l
    48a –> 42l
    48d –> 54a
    48l –> 43l
    49a –> 43d
    50a –> 44d
    51a –> 45d
    52a –> 46d
    53a –> 47d
    54a –> 48d

    • November 1, 2017 11:54 pm

      Fantastic, thank you! It will take me/us some time to catch up with this, likely until the Thanksgiving weekend.

      • Juh Raffe permalink
        November 14, 2017 12:08 pm

        Here’s a picture of the flex map:

        This doesn’t capture the details of the rotations and flips between flexes, but I find it easier to grasp the overall structure with a picture.

        The dot clusters are the 12 rotation/flip combinations of each state. The yellow and orange in the clusters are the rotations and flips between them (too small to see individual lines), and the pink lines are the twist flexes.

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