Part 1 History Calc Circ. Twist Tips Simple Twists Orbit Twist
Short History of the Atomihedron (9-2013) page 1B
The Atomihedron is an outgrowth of experiments with flexagons and flexahedrons. Martin Gardner wrote a popular column in Scientific American on mathematical recreations. In the May, 1958 issue he described hexaflexagons. These devices are made of twisted loops in the shape of a hexagon consisting of hinged equilateral triangle strips. The Hexaflexagons have a magical ability to turn inside out (flex) in a regular manner, almost organic, and hypnotic.
Hexaflexagons were invented in 1939 by Arthur H. Stone. Every so often someone devises new versions of these devices. My experiments began with linear hinged strips of right triangles about a year before I became aware of the hexaflexagons. These strips could be wound up in helix fashion but not flexed. Gardner's article started me experimenting with solid flexagons which I called flexahedrons. Several different articles were published on flexahedrons devices such as rings of regular tetrahedra, hybid flexahedrons, hexaflexatetrahedron, etc.
It was found that you could not make structures that always flexed inside out when you increased the number of links and increased the twist as much as possible. You had to have almost maximum twist or the structure would have no organized system to it. The twist removes most of the freedom and restricts it to organized movement only . This being the case, and with the current research at the time showing how DNA is twisted I theorized that twist is a very important self organizing principle.
Later it came to light that Louis Pasteur, with his discovery of enantiomorphic molecules came to regard asymmetry as a basic natural principle. In reality both symmetry and asymmetry have to work intimately together, at least in biology, and perhaps in physics as well.
I decided to experiment with a tetrahedron that fills space, is symmetrical and has two 90 degree solid angles and four 60 degree solid angles. Around 1964 I began experimenting with twisted loops. This involved a lot of head scratching to attempt to find the best possible configuration and correct twist. After many models the EH or Electrihedron, 24 right angled tetrahedrons hinged together into a loop, with two twists, was the only one that seemed to satisfy my requirements.
Further experimentation with the EH gave me the idea that the EH loops could be linked to each other. Of course when linked flexing is no longer possible except as a kind of time slicing system. From that time, about 1964 and on I have experimented with these linked EH circuits at odd times. In the early 70's I discovered the Atomihedron structure which can consist of many EH in linked and interwoven circuits as the order n, of AHn increases.
The Neutrohedron reverses the twist of the Atomihedron its Edual. Very recently I have found that these Eduals always fill space in combination with Eduals. This is a unique mathematical property but it has not yet been fully proven.
The Edual operation on the Neutrohedron reverses the twist of the Atomihdron and vice versa. Very recently I have found that these Eduals always fill space in combination with Eduals. This is a unique mathematical property but it has not yet been fully proven.
So far this system has not attracted a lot of attention since the mathematics of it has not been fully developed. As far as the names used, ie. Electrihedron, Atomihedron, Neutrohedron, Photohedron, these are fanciful and speculative since no direct connection with quantum theory has been proven. However I would point out that the system does many things that no other system does. It self organizes on many different levels. That alone should make it worth serious study. All this self organization comes about from ultimate simplicity, a single link, the EH. This link can be seen as a loop of 24 unit vectors.
This research started in the early 60's and continues in the present year 2013, representing over 50 years, on and off tackling this structure.
The Atomihedron is by itself very complex. As it goes to infinity the circuits become very complicated. Clusters of circuits form at rational fraction distances of the order n, from the edge to the center, with the biggest cluster in the center.
In the mid 80's I submitted a paper to The American Mathematical Monthly. They wanted to publish it but only if I could add more mathematics to the description. At the time I could not find a period of time where my attention would be solely turned to the Atomihedron so I never resubmitted the paper. Two other papers were published, both only minor portions of the EH system. One in "The Mathematics Teacher" in October, 1968 titled "Can Space Be Overtwisted?", the other in a student mathematics journal called "The Pentagon" Spring, 1972 titled "How a Flexible Tetrahedral Ring Became a Sphinxx". This paper Showed the order 2 Atomihedron and the order 2 Neutrohedron as well as the EH2 identity circuit for Eduality and the EH6 circuit. Higher order Atomihedra could have been illustrated but space was limited to 8 pages. These papers may eventually be included on this web site. I also self published two booklets, once called "A Philosophy of Twist" in the 60's and one titled "A Theory of Replication" in the 80's.
I made puzzles of the 6 link vesion of wood and plastic in the 80's. At present an injection mold has been made and is sold and is quite precise and fun to work with.
Any one wishing to work on this is welcome. I believe a program could reveal if it has more properties that emulate atomic particles. I can provide information if interested.
This system has a beautiful built in self organization property. Every time I come back to it and resume experimenting new discoveries come to light. It is very non intuitive. The way the links combine is always a mind game to figure out, even for figures that I have built many times. I have added solution videos on this website to aid anyone wanting to work these puzzles.