# Sonobe Cube Ratios and Construction

Last week’s hexagonal prisms of linear ratio 1:2:3 brought us to a discussion of dimensions of a cube. We looked at the building blocks of a cube: square faces, edges, and vertices, each a face, edge and vertex. The cube itself or any solid is three dimensions (we can use any three of the words “base, length, width, height, depth, or breadth” as long as they are on different planes. All solids are three dimensional (like the cube). The face of a solid is two dimensional (like the square face of a cube; this is the surface area). The edge or length from one vertex to another is one dimensional (like the segment of one edge of a cube; this is distance). Finally, a vertex or point has no dimension or zero dimensions (like the vertex of a cube; this is location).

Everyone likes to build solid structures so we explored origami cube constructions. The easiest, most reliable, and incredibly stable cube construction is the Sonobe Cube invented by Mitsunobu Sonobe. This requires six square pieces of paper and the steps are easy to replicate.

We focused on the most important element of the construction which is the creation of a square. Take any non-square rectangular piece of paper and construct an isosceles right triangle (a triangle with one right angle and two equal 45 degree angles with the two legs being equal length), and just pinch at one of the 45 degree vertices. Fold a rectangle at that point and you have a near perfect square. You can cut off that excess rectangle with a scissors, or by folding back and forth and slightly wet the fold and the tear is clean. Once you have six square pieces of paper, begin construction.

With the older Mathletes, we also explored the hypotenuse (longest side of a right triangle) of the isosceles right triangle and proved that it has a length of the leg multiplied by the square root of 2 or about 1.41. So if the isosceles right triangle leg is 1, the hypotenuse is root 2 or approximately 1.41; if the leg is 8”, the hypotenuse is 8 x root2 or approximately 11.28”.

As we folded, we noticed a 1:2 linear ratio rectangle, then  1:4 linear ratio rectangles, then we created our first isosceles right triangle folds. The Mathletes noticed that there were 32 of these isosceles right triangles in the large square. After you make the fold in step 6, the resulting isosceles right triangle in step 7’s picture is what ratio to the smaller isosceles right triangle in steps 4 and 5? The linear ratio is 2:1 and the area ratio is 4:1. The children could draw four small isosceles right triangles in the large. Step 10 forms a right trapezoid and asked the Mathletes how many small isosceles right triangles fit into the right trapezoid. There were 12 and sketching them is fun. Also in step 11, they can see that three large isosceles right triangles fit into the right trapezoid. Then the moment of truth comes at step 12 when it results in a quadrilateral. Most Mathletes impulsively say rhombus or diamond (first, diamond is not a mathematical figure). A rhombus has four equal sides. But usually one Mathlete could see that only opposite parallel sides were equal, but not all. So this is not a rhombus, it is a parallelogram.

Then we talked about parallel vs. perpendicular. The Sonobe Cube modules can only be put together with perpendicularity, not parallel. When they come in next week with their completed Sonobe Cubes, if all tabs are not locked in, I know they connected them without perpendicularity. Easy to fix.

Steps 13 and 14 create a medium sized isosceles right triangle. Four of these fit into the parallelogram, 2 fit into the small square, and 16 into the large square.

I created two 8-minute videos to show the children how to put them together. The first is called Sonobe Cube Construct Modules.MOV and the second is called Sonobe Cube Connect Modules.MOV. When you click on this link, the second video appears first so be sure to watch them in the proper order. It may take a minute to download each video so give it some time before you watch them. The children will need six square pieces of paper before they begin.