Introduction: A Method for Making a Cardboard Cube Showing Its Six Diagonal Planes of Symmetry
This Instructables provides one method for illustrating what geometric figure results when a cube is cut along its six diagonal planes of symmetry. Eight identical cubes are obtained when a cube is cut along the three planes of symmetry that are parallel to each pair of opposite faces. Of course these 8 cubes can be reassembled to reform the original cube. The analogous situation that occurs when a cube is cut along its six diagonal planes of symmetry is not so readily obvious.
The proposed method starts with the six rectangles which will end up being the diagonal planes of symmetry of the cube. Each of these rectangles has one side length equal the length of the cube’s edges while the other side has a length equal to the diagonal drawn on one of the cube’s faces. Using Pythagoras’ theorem, the length of the sides of the rectangles are in the ratio of 1 to √2.
The only materials needed to illustrate this construction are six different coloured sheets of card stock, scissors, clear sticky tape and a pencil or ballpoint pen. Four of the rectangles will be cut into smaller geometric figures and they will then be reassembled into rectangles as the model is made.
Once the model is constructed, some geometric properties of the cube that can be visualised on the constructed model are discussed.
Step 1: Cutting Card Stock to Make Rectangles and Drawing Lines on the Rectangles
Cut the six different coloured pieces of cardstock so as to form six rectangles whose sides are in the ratio of 1 to √2.
Draw on each rectangle both diagonals and a line parallel to the shorter side of the rectangle that bisects the rectangle’s longer sides. These lines mark where four of the rectangles will be cut into smaller geometric figures and where slits on all the rectangles will be cut. (Only half of the coloured rectangles with lines drawn on them are shown in the above photo.)
Step 2: Cutting Four of the Rectangles
Cut two of the rectangles along one of their diagonals so as to form four right-angled triangles. (Only one of the coloured cards which has been cut to form right-angled triangles is shown in the above photo.)
Cut another pair of rectangles along both diagonals so that each rectangle is divided into two pairs of two different isosceles triangles. Two of these pairs of isosceles triangles now have no lines drawn across them (unlike all the other geometric figures) and pairs of these unmarked triangles with different colours will be used to form two kites in Step 4 below. (Only half of the coloured rectangles which were cut into isosceles triangles are shown in the above photo.)
Step 3: Cutting Slits on the Various Geometric Figures Obtained in the Previous Two Steps
When cutting slits in the geometric figures, try and keep the width of the slits just under 1 mm.
On each of the two rectangles:
- cut four slits with each slit starting from a vertex of the rectangle and finishing one quarter of the way along the diagonal from where the cut began;
- cut one slit starting midway on the longer edge of the rectangle and finish the cut at a point half way along a line parallel to the shorter side.
On each of the four right-angled triangles:
- cut a slit which goes half way along a line which starts at the middle of the triangle’s longest side (the hypotenuse) and finishes half way along the line joining the mid point of the hypotenuse and the vertex of the triangle opposite the hypotenuse;
- cut a second slit along a line parallel to the shortest side of the triangle which starts at the mid point of the second longest side of the triangle and finishes at a point half way along the line parallel to the shortest side of the triangle;
On each of the four isosceles triangles which have as their longest side the longer side of the original rectangle from which they were cut (these four triangles will have a line drawn on them as a result of the lines drawn on the rectangles in Step 1), cut a slot starting from the vertex opposite the longest side of the triangle and finish the cut half way along a line joining this vertex to the middle of the longest side of the triangle.
No slits are cut on the four isosceles triangles without lines drawn on them.
Only one of each geometric figure with slits cut in them is shown in the above photo.
Step 4: Construction of Two Kites and Forming a Slit in Each Kite
Two kites are constructed from the remaining four isosceles triangles which have no lines drawn on them. Each kite is constructed from two of these triangles with each of the triangles forming the kite having a different colour.
Using clear tape, join together two of these isosceles triangles along their shorter sides (the sides on the original rectangle before it was cut) so as to form a kite. (Make sure the resulting figure is not a rhombus; opposite sides should not be parallel.) Repeat this construction for the other two isosceles triangles.
Cut slits along the line joining the two isosceles triangles starting at the vertex of each kite where the shorter sides meet and finish the cut half way along the length of the join.
Step 5: Creating Two Diagonal Planes of Symmetry
Using the two rectangles with the five slots, slide one rectangle into the other one along the slits parallel to the shorter sides of the rectangles, keeping the two rectangles at right angles to each other.
Step 6: Creating Four Diagonal Planes of Symmetry
Using two of the right angled triangles of the same colour, slide the longer slit on each triangle into a pair of opposite slits on one of the rectangles so as to reform the original rectangle made from the two right angled triangles.
Then using the other two right angled triangles, slide the longer slit on each of these triangles into either of the two remaining pairs of opposite slits available on one of the rectangles (note that two pairs of the four slits are unavailable on one of the pair of rectangles once the first reconstructed rectangle is made) again reforming the original rectangle made from these two right-angled triangles. When inserting the second pair of right angled triangles choose the surface of the triangle (either the one with or the one without lines drawn on it) so that the triangles do not get in the way of the first reconstructed rectangle.
The two rectangles that are formed from these four right-angled triangles share a common vertex with a pair of opposite vertices of one of the original rectangles which are at right angles to each other (i.e., one of the two rectangles that were slid into each other in the previous step).
Step 7: Creating Six Diagonal Planes of Symmetry; Part 1
Slide each of the four isosceles triangles with slits at their apices into the slits which appear on the longer sides of the reconstructed rectangles made from the four right angled triangles making sure that triangles of the same colour are placed on opposite faces of the nearly constructed cube. In this step each pair of isosceles triangles of the same colour is inserted into slits on the right angled triangles which also have the same colour. The photo shows two of the isosceles triangles inserted into slits using a similar orientation to the photo shown in the previous step with four planes of symmetry.
Step 8: Creating Six Diagonal Planes of Symmetry; Part 2
Bend the two halves of each of the kites along the joints that were taped together in Step 4 and slide the slit made on the kites into the opposite slits on the rectangle which has only one complete rectangle originating at its vertex. This is done in such a way that each of the colours of the kite’s triangles are in the same plane as the colour of the isosceles triangles inserted into the slits as described in the previous paragraph. Again the orientation of the photo is similar to that shown in the previous step.
Step 9: Geometric Figure That Results When a Cube Is Cut Along Its Six Diagonal Planes of Symmetry
Looking directly down on the faces of the cube, one sees that each face is divided into four right angled isosceles triangles. The sides of these isosceles triangles are formed from:
- the intersection of two planes of symmetry at right angles to each other (each of a different colour) with a face of the cube, and
- an edge of the cube.
Also looking down on the faces of the cube there are the other four diagonal planes of symmetry (each of a different colour and not the same colour as those mentioned in the previous paragraph) with each plane emanating from an edge of the cube.
Thus, including the face of the cube, each face has on its surface four identical adjacent tetrahedra. As three of the tetrahedron’s faces are at right angles to each other they are classified as trirectangular tetrahedra (a tetrahedron where all three faces at one vertex are at right angles to each other; two views of such a tetrahedron are shown in the above photos). One such tetrahedron is shown inserted into a face of the cube in the above photos. Such a trirectangular tetrahedron is an oblique pyramid (a pyramid where the apex of the pyramid is not over the centre of the base). Also of interest is the fact that a trirectangular tetrahedron provides a generalisation of Pythagoras’ theorem in three dimensions with sides of the triangle replaced by areas of the faces of the tetrahedron when stating the theorem.
Thus we conclude that the original cube could be reconstructed from 24 of these identical trirectangular tetrahedra.
The trirectangular tetrahedron shown in the picture was constructed by:
1. Noting the relative dimensions (relative to the length of the edge of the cube which is taken as 1 unit and using an actual length for the edge of the cube a couple of millimetres less than that used in constructing the planes of symmetry of the cube in order to account for the thickness of the cardboard stock) of the sides of the triangles that would make up the four faces of the tetrahedron;
2. Drawing the four triangles on a piece of cardboard;
3. Cutting out the four triangles;
4. Sticking the appropriate sides of the triangles together with clear tape to form the tetrahedron.
For this trirectangular tetrahedron the relative dimensions of the sides of the four triangles are:
1. Triangle on the face of the cube: 1, √2/2, √2/2;
2. Triangle on the diagonal plane viewed when looking directly down on the face of the cube: 1, √3/2, √3/2;
3. Two triangles on diagonal planes at right angles to the face of the cube: 1/2, √2/2, √3/2.
Step 10: Step 10: a Further Observation
Imagine that the three planes of symmetry that are parallel to opposite faces of the cube were also present in the constructed cube (looking at one face of the cube, as well as seeing the diagonal planes that are at right angles to each other on the cube’s face, one also sees two planes at right angles to each other bisecting these diagonal planes). Then there would be eight r tetrahedra making up each face and 48 such tetrahedra could be used to construct a cube. However these tetrahedra while congruent would not be identical. 24 pairs of these tetrahedra would be mirror images of the other 24 tetrahedra. The above photo shows one such tetrahedron inserted into one of the cube’s faces. The tetrahedron was constructed in a similar way to that described in the previous step.