Flexagon
Flexagons are articulated polygons with the property, after a folding manipulation, called pinch-flexing , to reveal further sides in addition to their front and rear sides.
history
The first model of a flexagon was discovered in 1939 by the British student Arthur H. Stone at Princeton University : he was a British exchange student in the USA at the time. Since the American paper format is larger than the DIN A4 format used in Europe , Stone had to remove a strip of the paper to make the new paper suitable for his folder. Out of boredom in lectures, he played with these strips, folding them into various shapes. One of his results was what was probably the first Trihexaflexagon. The aforementioned folding property of this paper model prompted him and his fellow students John W. Tuckey , Bryant Tuckerman and Richard Feynman to found a Flexagon Committee and to coined the name 'Flexagon' for this type of object. In 1956 they were introduced to a wider audience through Martin Gardner's article in The Scientific American . Furthermore, they discovered and described important analysis procedures such as the Tuckerman traverse and the associated Feynman diagram . A complete mathematical theory of flexagons (English: Flexigation ) was worked out by Tuckey and Feynman in 1940, which also contains instructions for the construction of any flexagon. The theory was never published, but bit by bit it was rediscovered by other mathematicians.
However, there is no solid evidence that the flexagons discovery took place or that this report ever actually existed. One could e.g. B. object that the first flexagon, the Trihexaflexagon, is a very simple Möbius strip , which was described as early as 1858, of which there were apparently reports from Vienna in the prewar period. Nevertheless, the Princeton Committee was the starting signal for work on flexagons: The Princeton Flexagon Committee also includes Briant's father, Prof. Louis B. Tuckerman, who continued after the committee was dissolved as a result of the war (especially the attack on Pearl Habor ) presented a Flexagon to the winners of the Westinghouse Science Talent Search every year . This type of dissemination led to the first report on flexagons. In 1956, Flexagone was introduced to the general public by Martin Gardner's first contribution to his " Mathematical Games " column in Scientific American . The first report from Oakley and Wisner followed almost at the same time, going beyond the descriptive level and mathematically analyzing flexagons. A small variety of other descriptive articles followed until 1962 when Conrad and Hartline's RIAS report on flexagons was published. This not only corrects the mistakes made in Oakley and Wisner's report, but also provides instructions on how to construct any flexagon.
Types of manipulation
Flexagons or hexaflexagons are usually manipulated with the pinch-flex. With higher-order flexagons, however, another type of manipulation, the V-Flex, is possible. Square flexagons, i.e. flexagons with a square basic shape, are folded using as yet unnamed types of manipulation that resemble the movement of opening a book.
Instructions for building a Trihexaflexagon
1. A Trihexaflexagon is formed from 9 equilateral triangles, which are drawn on a strip as shown in Figure 1.
2. The triangles are marked on the front and back as shown in Figure 1.
The strip pictured is referred to as the flexagon mesh. The strip was turned along the horizontal axis for labeling. The first triangle on the left is marked with the number 2 on its front and the number 1 on its back.
3. The lines between the triangles are folded.
4. The strip is labeled in such a way that two adjacent triangles bear the number 3 in three places (also note the back). These neighboring triangles are folded against each other and thus made to coincide. Figure 2 shows how the first fold was made on the left.
5. None of the triangles labeled 3 are now visible.
6. The end edges colored green in Figures 1 and 2 are joined together, i. H. glued z. B. with scotch tape. This results in a hexagon.
7. The Trihexaflexagon is now ready (see Figure 3).
nomenclature
When looking at a Trihexaflexagon, built from a face-oriented network, it is noticeable that all six sides of the triangle are labeled with the number 1 on the front. All six sides of the triangle are labeled with the number 2 on the back. The flexagon was formed from nine triangles, so it consists of eighteen triangle sides (front and back). The flexagon shows only 2 * 6 = 12 triangle sides; 6 triangle sides are covered, namely all triangle sides that are labeled with the number 3.
Regarding the nomenclature: The nine triangles are called triangular leaves or generally leaves and the eighteen sides are called triangle sides or generally sides .
The front with six sides, all marked with the number "1", and the back with all sides marked with the number "2", are faces of the flexagon. The face of the front is called the front face in this work. The mesh in Figure 1 has been labeled so that the sides of a face are all labeled the same. This type of network is called a face-oriented network.
As Figure 3 suggests, the front face is not a regular, continuous hexagon, but is always structured by a gap after 2 sheets, i.e. a total of 3 times. If you look between the columns, you can see the hidden 2 · 3 = 6 sides.
If you free the flat flexagon from its two-dimensional world by opening the gaps a little, you can see that a Möbius strip was created by bringing the strip together. A simply twisted Möbius strip is made from a strip by turning the strip 180 ° at one end and connecting the ends. When building the Trihexaflexagon, the strip was turned three times by 180 ° by folding. The face of the flexagon shows six sides. If you don't just want to capture the pages, but also consider all sheets of the flexagon, the introduction of the term sheet bundle or English: Pat is necessary. The flexagon consists of six bundles of sheets which together encompass all sheets and thus also all sides of the flexagon.
The sheet bundle from location A contains 2 sheets with 4 sides. The top sheet has a visible side that is part of the front face and is marked 1 and a back side that is marked 3. The bottom sheet has a visible side marked 2 as part of the back face and a hidden side marked 3. The bundle of leaves from location B consists of only one leaf, both sides of which are both visible as part of the faces. The number of sheets in a sheet bundle is equal to the order of the sheet bundle. As a result, the bundle of leaves at position A is of order 2, the bundle of leaves at position B is of order 1.
The leaf bundles from location C and E correspond to those from location A. The leaf bundles from location D and F correspond to those from location B. Thus the AB sequence is repeated three times in the flexagon. Therefore one can adequately describe a flexagon with two adjacent bundles of leaves. Two adjacent bundles of leaves are called a sector (see Figure 6). Adding the order of the two bundles of leaves of a sector gives the order of the flexagon. The Trihexaflexagon is therefore a 3rd order flexagon.
Naming of flexagons
Flexagons are named for their number of faces and their shape. The first prefix indicates the face number, the second the number of corners of the shape. A Trihexaflexagon 3 = Tri has different faces and represents a hexagon = hexagon. When counting the faces, it should be noted that only the faces that have been created with a pinch flex are counted. Other forms of manipulation are possible, which also form other faces, but these are not taken into account.
A tetrahexaflexagon has four faces and a hexagon as a geometric shape. The hexasquareflexagon has 6 faces and a square as its basic geometric shape.
Variations of flexagons
Hexaflexagon
The group of hexaflexagons includes all hexagonal flexagons. The most common and famous representatives are built from equilateral triangles, like the first Trihexaflexagon by Arthur H. Stone .
Trihexaflexagon: This flexagon consists of 9 equilateral triangles. There is a variation.
Tetrehexaflexagon: This flexagon consists of 12 equilateral triangles. There is a variation.
Pentahexaflexagon: This flexagon consists of 15 equilateral triangles. There is a variation.
Hexahexaflexagon: This flexagon consists of 18 equilateral triangles. There are three variations.
Heptahexaflexagon: This flexagon consists of 21 equilateral triangles. There are four variations.
Tetraflexagon (folding square)
Examples are Tritetraflexagon and Tetratetraflexagon .
Web links
supporting documents
- ^ Pook, L .: (2003). Flexagons Inside Out, Cambridge: Cambridge University Press
- ^ McIntosh, HV: (2000). My Flexagon Experiences, Puebla, Mexico: Departamento de Aplicación de Microcomputadoras, Instituto de Ciencias, Universidad Autononoma de Puebla.
- ^ Gardner, M .: (1959). The Scientific American Book of Mathematical Puzzles and Diversions, New York: Simon and Schuster
- ↑ McLean, B .: (1979). V-Flexing the Hexahexaflexagon, The American Mathematical Monthly, Vol. 86, no. 6th
- ↑ Oakley, CO; Wisner, RJ: (1957). Flexagons, Haverford College: The American Mathematical Monthly, Vol. 64, No. 3
- ↑ Bielefeld University - Flexagone - available online at: Flexagone. last accessed on February 12, 2016
- ↑ Oakley, CO; Wisner, RJ: (1957). Flexagons, Haverford College: The American Mathematical Monthly, Vol. 64, No. 3, p. 143