Tetraxis®, a puzzle by Jane and John Kostick
The Tetraxis® toy is a geometric puzzle. It is made of twelve identical pieces. Several magnets help guide and hold the pieces together and it is a puzzle not difficult to solve and a real pleasure. It is not difficult to build one using cardboard and some magnets or a 3D printer.
Tetraxis® is a registered trademark of KO Sticks LLC.
On this page we are going to study some of its properties, in particular, the relation between the Tetraxis® toy and the rhombic dodecahedron, the shape of the hollow space surrounded by the twelve pieces of the puzzle, which you can build by downloading a template.
The name Tetraxis® means four axis. The Greek prefix 'tetra' means 'four' (as in 'tetrahedron', four faces). This is because the twelve pieces of this puzzle line up along four axes.
These four axes are related to the cube. The twelve pieces line up in the directions of the four long diagonals of a cube.
We also encounter these long diagonals when studying a polyhedron discovered by Johannes Kepler: the rhombic dodecahedron.
A rhombic dodecahedron is a polyhedron that has twelve rhombic faces. It can be seen as a cube capped with six pyramids:
The rhombic dodecahedron has wonderful properties. It tessellates space in a way that is related to how bees build their honeycombs. It has vertices of two different kinds: four edges met in six of these vertices of three edges met in eight vertices of the rhombic dodecahedron.
Studying the relationships between the Tetraxis® toy and the rhombic dodecahedron it is interesting to pointing out the ratio between the two diagonals of the rhombic faces of this polyhedron.
We could calculate the diagonal D of one of these rhombuses:
These rhombuses are related with the proportions of the standard size of paper DinA.
TO LEARN MORE ABOUT THE RHOMBIC DODECAHEDRON
I built this model to have a better understanding of Tetraxis®. It is made up several rhombic dodecahedra:
This structure is inspired in this beautiful video.
These are the four directions on a rhombic dodecahedron:
You can build a Tetraxis® using cardboard. Some magnets inside hold the pieces together. Neodymium magnets are perfect but they could be dangerous for children (in particular, if they eat them).
You could download this template, print it on cardboard, cut, fold and glue it to build the shape of a Tetraxis® stick. We need twelve pieces and we can use four different colors.
Circles indicate the location of the magnets. You can glue four magnets (notice the two different polarities). I used two neodymium magnets and two pieces of metal. Magnets and pieces of metal will remain inside the stick.
As an alternative, using the following model you can build a smaller Tetraxis®. I think that the big one is much better!
This is the result. A wonderful puzzle, easy to build and a pleasure to solve it.
There is another different way to draw the plane net of the pieces of the Tetraxis® puzzle. The following template is inspired by an original drawing by Jane Kostick:
Nine small rhombuses make a similar big rhombus that is the exterior shape of each piece. Four magnets are at the centers of four of the smaller rhombuses.
The diagonals of these rhombuses are in the ratio square root of two.
More about the geometry of the Tetraxis in Kosticks web site.
Another way to build a Tetraxis® is using a 3D printer.
In the following photo we can see how the four colors meet in some places.
There are six of this combinations of four pieces. These are related with the six faces of a cube or the six vertices of a rhombic dodecahedron where four edges met.
In the following photo we see a combination of three pieces that is like a bow tie. There are twelve of these, one for each piece of the puzzle. Each of them is related to one face of a rhombic dodecahedron or to one edge of a cube.
There are eight combinations of three colors like the following one. Each is related with the vertices of a cube or the eight vertices of a rhombic dodecahedron where three edges met.
More about these combinatorial properties of the Tetraxis in Koskticks web site.
The shape of the hollow space inside an assembled Tetraxis® toy is a rhombic dodecahedron.
A beautiful video about the Tetraxis and the rhombic dodecahedron: Dissecting a Rhombic Dodecahedron with Miles.
I would like to thank Jane and John Kostick for their kindness and generosity.
The Tetraxis and the rhombic dodecahedron (Dissecting a Rhombic Dodecahedron with Miles). A webpage in Kostick's site about dissecting a rhombic dodecahedron with a wonderful video.
Article by George Hart about the Tetraxis. There is a large Tetraxis in the Museum of Mathematics in New York.
Math In Your World. Solid Geometry: Wood Scuptures by Kosticks. Article by Ken Fan. Reprinted from The 'Girls' Angle Bulletin, volume 7, pp. 19,23-26.
Johannes Kepler - The Six Cornered Snowflake: a New Year's gif - Paul Dry Books, Philadelphia, Pennsylvania, 2010. English translation of Kepler's book 'De Nive Sexangula'. With notes by Owen Gingerich and Guillermo Bleichmar and illustrations by the spanish mathematician Capi Corrales Rodrigáñez.
Hugo Steinhaus - Mathematical Snapshots - Oxford University Press - Third Edition.
Magnus Wenninger - 'Polyhedron Models', Cambridge University Press.
H.Martin Cundy and A.P. Rollet, 'Mathematical Models', Oxford University Press, Second Edition, 1961.