## Colin McAllister

### Templates for Asymmetric Integer Tetrahedra

**Image: An Asymmetric
Tetrahedron with Prime Numbered Edge Lengths**

Article 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. Theycould 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+Tetrahedron*

There 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.htm*

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