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Trailblazing Flexagons - Findings
Some Major Flexagon Site Links:
My Trihexaflexagon Version * Mine vs Traditional
My 'Magic' Flexagon
Flexagon Background * Many References
Faces & Strips/Free Downloads

'Traditional' Version
Assembly & Flexing With Excellent Directions!
Mathematische Bastelien

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General Trailblazing Flexagons Site Map
Trailblazing Flexagons Visual Arts Site Map
Vern's Visual Victuals - Personal Digital
and Fine Art Creations

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'Weak'ly Wit - Vern's 'Pun'derosa
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Trailblazing Hexa-Dodeca-Flexagons?
Go to Flexagon Discovery: The Shape-Shifting 12-Gon for details on the hexa-dodeca-flexagon, a newly discovered hexagonal flexagon with 12 pats and 6 faces that is folded from a straight strip of paper. This flexagon displays some unique characteristics, including these: it flexes into a new, flat shape for some of its flexes and it naturally mixes leaves from different faces to form hybrid faces.

More Flexagon Fever at

EighthSquare.com

Trailblazing Flexagons

Findings/Summary


New! Holographic Flexagons!


Flexagon Display
City Library St. Cloud, MN
October 2004

Wide Variety of Flexagon Designs, Plus, New And Old Flexagon Phenomena Featuring Trailblazing Trihexaflexagons!

This Page Revised 23 Mar 2005
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Below Findings/Overall Summary Topics
Version Comparison - Math - Art - Music - Philosophy
Psychology - Word Play - Origami - Flexagon Chains - Miscellaneous

VERSION COMPARISON: The traditional trihexaflexagon version and my version have some interesting similarities and differences. The traditional version is constructed by twisting the band three half twists before connecting and mine is only twisted once. When the traditional one is cut down the middle, it produces one twisted band looped inside another twisted band while my version produces one larger twisted band.

The traditional version has two adjacent triangles before being interrupted by a fold connecting another two adjacent triangles on each face. My version has three adjacent triangles interrupted by two adjacent triangles and then one triangle on each face. Manipulation procedures and their respective difficulties are different for each version. Some different designs cannot be reproduced between the two after converting one version to the other.

MATH: At this time, the following different number sequences and varying the positioning of these on the two flexagon versions are being explored: 1 - 6, 1 - 9, 1 - 12, 1 - 36, 1 - 54, Fibonacci and Pascal’s. Furthermore, different mathematical operations on the numbers like squaring and cubing are being tried in the search for patterns.

I find that my 'Magic' Flexagons are a great help when explaining to people that a Möbius Band has only one side and one edge. The repeat of the same color number on opposite sides of the band, plus, the same numbered triangles on opposite sides of the the band helps in explaining the one side and one edge attributes of the Möbius Band. Furthermore, the similarity to a 3X3 magic square which adds to 15 in each row, column, and diagonal is interestingly present in this trihexaflexagon. Each triangle adds to fifteen. Each face adds to 90 and each of the three pairs add to 180.

Considering that anything goes for a 'try'hexaflexagon, sequences were graphed using SPSS and SyStat producing some interesting results (curve fitting - cubic formed a repeating sine wave on the 1 - 12 sequence version).

A future study will entail cutting the flexagon down the middle and also a third of the way to see what patterns and phenomena occur when the resulting larger band or the small band within larger band are assigned all sorts of sequences. The Sierpinski Triangle is also being tested for possibilities. It would be interesting to find an assignment of numbers that would produce a 'magic' flexagon similar to the one resulting from the numbering the side of the band 1 - 9.

Other future explorations will include more topology plus combinatorics such as the Fano Plane. The math is sure beyond me, but, it does look like venturing into things like quotient spaces and maps is worth a thought or two.

Hmmmmmm. Could it possibly be that Fermat trimmed the margins of books where he had written notes to discard what he felt were errors? Did his frustrated fiddling finally flexagon phenomena? Is there a yet to be found 'Fermat'agon lurking somewhere? I wonder if Fermat had a comparable 'magic' flexagon's last theorem hidden somewhere in an obscure book margin?

ART: Computer graphics now has another wide open window for creativity with the establishment of templates clearly depicting the flat band and the final faces for both configurations. Different materials such as transparent surfaces that may be used with printers will provide capabilities for an infinite number of designs. VELCRO furnishes another fascinating versatile factor to the fun. Minus the computer technology, one finds still another expanded variety of horizons that were not in sight before the addition of the new configuration.

MUSIC: Different scales, chords, and arpeggios have been tried with no real success in that area yet. If one looks at only the Moebius band aspect and takes the introduction to The Thunderer March by Johh Philip Sousa, Shepard scales would have an excellent continuing version of the nine measure introduction. One 9-note version of the blues scale could be assigned.

PHILOSOPHY: The Möbius Principle can be worked with and then there are many other philosophies that also may be considered such as the Tao Te Ching (81 verses could be split into nine triangles). Synchronicity may be simulated with the overlapping of triangles. My own 'Flex'osophy is starting to form ('band'ing around "Adopt adapt to become adept"). I hope to have by fall of 2003 to have a personal 'Theory for Everything' 'Flex'osophy on this page.

PSYCHOLOGY: Mandalas, and just recently kaleidoscopes, have been found to be therapeutic. Since the flexagon is somewhere in between these two, it likely can be added to the long list of alternative medicines that are now available. C. J. Jung attributed his painting of mandalas as helping him overcome his deep depression. Who knows. Maybe if he would have know about flexagons, he would have transferred the mandala concept to flexagons. The result could have been his adding of the flexagon to his book, "Man and His Symbols."

WORD PLAY: Palindromes are relatively much simpler to apply to flexagons than anagrams. Here is where a little computer assistance might be necessary in order to find some possibilities that I certainly believe exist. Of course anagram experts would probably be able to rattle off hundreds once the patterns become familiar. Crossword puzzles also seem to have potential. These could get quite complex or remain fairly simple. I’m sure that other word games could be designed.

ORIGAMI: Large flexagons and flexagons within flexagons could be designed. Even flexagon 'quilts' could be assembled by interlinking any desired type or design of flexagon. As the new possibilities unfold from the added capability for switching between the two versions, a whole new category of origami may be created.

The traditional flexagon is normally designed in such a way that the three different faces show up individually. However, by rotating by 180 degrees the three sets of triangles comprising one of the faces, it is possible to produce a situation with both sides having a face. This is excellent for applications like mobiles, tree ornaments, etc.

FLEXAGON CHAINS: Linking flexagons or combinations of flexagons and other possibilities opens even more fields to investigate. There could be applications in constructing models to help understand molecules performing as if they either were one-sided surfaces or if they included one-sided surfaces. Designing molecules could be made easier having the imagination assisted by physical flexagon models.

MISCELLANEOUS: The flexagon color wheel surprised me with its cooperation at times! How about Braille flexagons? It could be possible that more efficient circuits than existing circuits may be designed where the different flexagon versions and the different flexes produce different routes. It’s conceivable that many different types of games could be devised with the ‘cloaking device’ flexibility. The flexagons could be pieces for some sort of board games or just used themselves for traveling type games. Since there are 54 triangle edges, a regular card deck with two jokers could be applied somehow. A whole field of flexagon chemistry could be explored - flexorganics and flexinorganics for starters.

Mathematical discoveries often have more to say than their discoverers had ever intended or anticipated. (Ivars Peterson)

The reason that Einstein caught all scientists off-guard was that Einstein was what Bridgman called 'operational;' that is, he paid complete attention to, and interconsidered all the circumstances surrounding any scientific discovery. He did not isolate the discovery, but paid attention to all the circumstances of its occurence. (R. B. Fuller)

Re'flex'ions are welcome!

Verndrei@aol.com
Links to Major Sections:
General Trailblazing Flexagons Site Map
Trailblazing Flexagons Visual Arts Site Map
Vern's Visual Victuals - Personal Digital and Fine Art Creations
'Weak'ly Wit - Vern's 'Pun'derosa
Vern's Views - Opinions and Musings
Invisible Chronic Illnesses - Information and Analyses
Vern's Home Page