The 4colour cubeTuesday, 30 May 2017  Paul Stephenson An unmarked regular tetrahedron has planes of symmetry. It does not therefore have left and righthanded forms. (The technical word for handedness is chirality.) But here are nets for two tetrahedra in which each face has a different colour. Focus on the blueyellow rhombus. In one net green lies on the left, red on the right; in the other, the order is reversed. Alternatively, focus on the central blue triangle. In one net the surrounding triangles run redyellowgreen clockwise; in the other, they do so anticlockwise. And we can analyse the completed tetrahedra in the same way. The next picture shows a cube in which each corner has a different colour. One of the corners is hinged open and we see how a tetrahedron fits inside. And of course there must be two such cubes: a lefthanded and a righthanded. These are the '4colour cubes' of the title. Here are nets for the two 4colour cubes: In our touring maths lab, The Magic Mathworks Travelling Circus, we have a set of 64 wooden 4colour cubes: 32 lefthanded, 32 righthanded. In the picture below 1 is a 'cubeofcubes'. There are 8. Cubes of opposite chirality share a face so there are 4 of each type. The completed cube has red corners. In 2 we separate the top and bottom layers and split each into two pairs. In 3 we flip each pair over so that what was on the bottom of each cube is now on the top. In 4 we reassemble the cubeofcubes. It now has blue corners. Proceeding in this way, the experimenters can complete two cubesofcubes of each corner colour and assemble all 8 into a 'cubeofcubesofcubes': From the pile of cubes all jumbled together give one person a single lefthanded cube and the other a single righthanded cube. They must race to complete these two diagonallystriped walls: The same wall seen from both sides. However, the pattern running along the top and down the edges differs in two cases. The fact is that one is made from lefthanded cubes, the other from righthanded ones. Here is a pair of 4colour cubes in 'solid' Polydron: If you're a teacher, I recommend you obtain enough right angle triangles (2 bulk sets) for the children to build 4 of each so that the class can do the cornerswapping exercise I described.  Your article on our blogHave you written an article that you would like us to post on our blog? Is so then please send us your article for review.

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