# A New Twist on Flexagons?

* For Martin Gardner’s 100th birthday *

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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 in a calculus context never appeared in them.

Yesterday was the 100 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 given enough values for It also led to my favorite “visual proof” that : For any integer , if you write out the powers going across and take differences of adjacent values repeatedly to make an infinite equilateral triangle pointing down, the left side has the powers of . Iterating this gives you the powers of , but the entry for as counts down to steadfastly remains .

## 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 ” 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.]

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)

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):

and the last slide before the references

https://www.academia.edu/7108129/The_Four-Color_Map_Theorem_Kempe_s_Fallacious_Proof_Repaired_

These are beautiful, thank you!

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: http://youtu.be/F5cj3LL8pqE 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.

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…

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

http://www.khanacademy.org/math/recreational-math/vi-hart/hexaflexagons/v/hexaflexagons

http://www.khanacademy.org/math/recreational-math/vi-hart/hexaflexagons/v/hexaflexagons-2

Paul.

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 to integers, either non-negative integers or positive integers 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 and finite partial functions from to All of this comes straight out of the primes factorizations. That type of bijection may remind some people of Dana Scott’s Corresponding to the positive integers there arose two species of graphical structures, which I dubbed

riffsandrotes.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

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

Here’s a picture of the flex map: https://imgur.com/a/NE3zD

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.