Templates for Asymmetric Integer Tetrahedra

Templates for Asymmetric Integer Tetrahedra

Image: An Asymmetric Tetrahedron with Prime Numbered Edge LengthsArticle and templates by Colin McAllister, June 2015

http://cmcallister.typepad.com/blog/2015/06/templates-for-asymmetric-integer-tetrahedra.html

We are all familiar with the Regular Tetrahedron. It has six edges of equal length and four faces that are equilateral triangles. An Asymmetric Tetrahedron has edges of different length, and triangular faces of different shape and size. Asymmetric Tetrahedra are interesting because of symmetry and the combinations in which the edges can be arranged to construct a three dimensional solid. An Asymmetric Tetrahedron has no symmetry of its own, but it does have a mirror image solid that is otherwise identical, like its reflection in a mirror. This has applications in Chiral Chemistry, e.g. in a molecule where bonded atoms define the four different corners of the tetrahedron.

There are various ways to make a tetrahedron. You can construct one from six rods attached at their end points, or from four triangular sheets. I use four triangles drawn on a single sheet of paper. I apply the method of Compass and Straightedge Construction to create matching triangles. I use the free Geogebra software. Circles are drawn with radii equal to the edge lengths, and the corners of the triangles are defined by the intersections of these circles. I set the circles as hidden before printing the template. Only the outlines of the four triangles are printed. You don't need a computer to do this. You could easily draw the templates on a sheet of paper or card using a pair of compasses and a ruler. Experiment to find the best fit of your construction to the page, for minimum waste.

I became interested in asymmetric tetrahedra that have integer length edges. When you define limitations on a mathematics problem, it often makes the problem more interesting. The relative edge lengths can be sequential, e.g. 2, 3, 4, 5, 6 and 7. There are four different tetrahedra with these edge lengths. I drew templates for two of them. They are distinct, containing triangles of side 4,5,7 and 4,6,7 respectively. The other two are mirror image solids, which can be made by folding the templates inside out. The tetrahedron containing the 4,6,7 triangle is drawn as two pieces to best fit on an A4 or US Letter size page. I separated the 3,5,7 triangle and rotated it about 90 degrees to the right. You may experiment with scaling to print the largest template that fits on a page, or set to inch scale so that the edges can be easily measured. To save ink or toner during printing, don't fill the triangles with colour. I haven't printed any of these templates yet. It is clear that they will fold into tetrahedrons from the arrangement of the constructive circles, which link edges of equal length.

My ideas about asymmetric tetrahedra have very little novelty. The related topic of Integer Triangles has been studied for millennia. They could easily be constructed from lengths of string with equidistant knots, i.e. Stone Age technology.

Application of such shapes is unlimited. The proportions of the 2, 3, 4, 5, 6, 7 tetrahedron would make a tent for camping, with the largest triangle being the groundsheet and the smallest triangle as the door flap. It could be framed from six 2 foot poles and five 3 foot poles. The outer fly-sheet could be made as another tetrahedron by adding one foot to each edge length.

I wondered if a tetrahedron could be constructed from edges with lengths that are sequential prime numbers. The image above shows a paper template that folds into a tetrahedron that has relative edge lengths which are prime numbers: 3, 5, 7, 11, 13 and 17. The smallest face (filled black) is a triangle of sides 3, 5, and 7. When folded, this triangle meets the dark blue and light grey triangles to form a tetrahedron. The light grey and dark blue triangles fold, to meet along the edge of length 13. (The dark blue triangle is the upper one in the diagram.) Before cutting out the template, draw tabs around the perimeter edges, for gluing the paper shape together.This is the smallest asymmetric tetrahedron from the sequence of prime numbers, and is only possible in one configuration, plus its mirror image. If the smallest triangle is the base, the apex would not be above the base. If the centre of gravity is not above the base, it would fall over. If using prime lengthed edges, the centimetre unit is convenient for templates that will fit on a page. You could also make this tetrahedron from drinking straws with prime numbered length, cutting the shortest straw to 3/17th of full length. Not all sequences of prime numbers can be used to define a tetrahedron. The three primes 2,3,5 cannot form the sides of a triangle, so there is no tetrahedron with edge lengths 2,3,5,7,11,13. There is no triangle with side lengths that are numbers from the Fibonacci sequence. Another sequence that works is Numbers n such that 2n-1 is prime, https://oeis.org/A006254. I didn't discover any magic properties arising from using prime numbered edge lengths. You could use any system of your own to select edge lengths, and they don't have to be integers.

The templates are free to download from the geometry resource site I2Geo.net. http://i2geo.net/xwiki/bin/view/Search/Simple?terms=Asymmetric+TetrahedronThere are six files, three Geogebra files and three corresponding PNG image files. Load the 'ggb' files in Geogebra to see the circles that I used to construct the triangles. The files are my own work and I release them as public domain. One way to print the files is to insert the PNG images into a word processing document, scaling the image and minimising margins for efficient use of the paper.

Further development is limited only by your students' imaginations. There may be applications too. I suggest tiling of a surface, such as the wall of an anechoic chamber, with a random pattern of asymmetric tetrahedra. How could these shapes be used as building blocks in a three-dimensional construction? An abstract animal-like bilaterally symmetric shape could be assembled from the tetrahedrons and their mirror images. Another method of space filling is that sheet-based and rod-based tetrahedra could be mixed. Make a core asymmetric tetrahedron from triangular sheets. Then surround it by its four mirror images, made out of drinking straws. Only 12 rods would be required, as the core provides four base triangles. These are only suggestions. There are no rules, except those you make yourself, and those you discover from the inherent mathematics.

I was inspired to create these geometry templates by Dr. Diana S. Perdue's discussion Request for applications on a math problem in the LinkedIn group Math, Math Education, Math Culture.I would also like to thank Dr. Linda Fahlberg-Stojanovska for advising that the templates would be useful for mathematics classes, and for suggesting improvements to their design.

Here are two related links for advanced reading:

The Heronian Tetrahedron: http://mathworld.wolfram.com/HeronianTetrahedron.html

Tetrahedra with Edges in Arithmetic Progression: http://www.mathpages.com/home/kmath665/kmath665.htm15 Jun 2015 20:36:39