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Slicing in Different Dimensions

When Froebel presented his geometric gifts, he did not want them to appear static. One of the first gifts was a display of three basic forms suspended by strings in various ways (Figure 35). As the objects rotated, children could observe them from different views and ultimately come to an appreciation of their symmetries and structures.

Figure 35. Froebel's kindergarten included basic shapes that could be hung from eyelets at different positions, then viewed from different perspectives to see various cross-sectional shapes.

In the model devised by Froebel, the sphere, the cylinder, and the cube all had eyelets attached so that they could be suspended in different ways. Because of its symmetry, the sphere had only one eyelet. The cylinder had three: one in the center of an end disc, one in the center of a side, and one on the rim. The cube also had three: one in the center of a face, one in the center of an edge, and one at a vertex.

The various views of these rotating objects lead to one of the most intriguing exercises in understanding forms in space, namely the determination of cross-sectional slices. One way to visualize this without actually applying a knife to a real model is to imagine what would happen if we gradually submerged the block in water. How will the shape of the water level change?

Figure 36. The central diagonal cross section of a cube turns out to be a regular hexagon whose six edges cut off triangles on each of the six faces of the cube.

The exercise that is most difficult for students is to visualize the shape of the "equator" of a cube suspended from a vertex. A student who has looked carefully at a real cube will have a much better chance of figuring out that the answer is a hexagon (Figure 36). This fact can be demonstrated nicely by stretching a rubber band around a cube. A cardboard model for the pieces of this decomposition of a cube can be made by cutting corners from three squares and placing them on the sides of a regular hexagon (Figure 37).

Figure 37. By folding this template into a solid figure, one gets half of a cube sliced on the central diagonal. Two such solids can be reassembled to form the cube by placing the hexagon faces together.

A transparent plastic cube half filled with a colored fluid can be manipulated to show the various slices through the center. If the cube is exactly half full, the shape of the liquid's surface will always be a central slice — that is, a slice through the center — regardless of the cube's orientation. It is a good challenge to then ask students to figure out which position of the cube produces the central slice with the greatest area. (It is not the hexagonal slice!)

Already in the last century when Milton Bradley took up the manufacture of Froebel's kindergarten materials in the United States, he included in one of his sets another figure-a cone. The conic sections are phenomena that can be seen and appreciated long before students are introduced to analytic geometry. Once again, a transparent cone partially filled with liquid can illustrate the changing conic sections as the object rotates.

Figure 38. As the central slice of a six-sided cube yields a regular six-sided polygon, so the central slice of a four-sided tetrahedron yields a regular four-sided polygon-that is, a square. The template on the right proves the means for constructing half of a tetrahedron; two such pieces make an excellent geometric puzzle.

Figure 39. Appearances can be deceiving: the direction of the arrowheads changes the apparent length of the lines without changing their actual length.

The investigation of slices of polyhedral objects leads to an interesting puzzle. If we slice a triangular pyramid by a plane parallel to one of its faces, we get a series of triangles. If we slice by planes parallel to one of the edges, we get rectangles, and in the central position, a square (Figure 38). Students can make cardboard polyhedral models of the two pieces of this decomposition by cutting and folding an appropriate pattern. Many people find it very difficult to put these two identical pieces together to form a triangular pyramid. The difficulty seems to be a three-dimensional analogue of the optical illusion that makes two lines of equal length seem different if we put arrows on the ends (Figure 39). [an error occurred while processing this directive]