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Drawing Cubes

All children in Froebel's kindergarten practiced drawing. They played with drawing on one level while they learned on another. They learned to observe spheres, cylinders, and cubes; ultimately they learned to draw what they saw. In our day there is not as much emphasis on drawing, so we miss opportunities to develop the ability of our students to visualize geometrical relationships.

The most common way of representing a cube in most books is to draw a square, then translate it along an oblique axis (usually at a 45° inclination) and then to connect corresponding points (Figure 21). Although this is a perfectly valid representation of the structure of a transparent cube, no view of a cube actually looks like this image. Whenever we look at a cube, if one face appears as a square, then we must be looking directly toward that face; in this case the opposite face will be directly behind the face we see and not off to the side as it is in the traditional drawing. This is true whether we use a straight-down "orthographic" projection or foreshortening (Figure 22), where the back face appears smaller than the front.

Another popular method of drawing uses "isometric projection," which expresses three edges of a cube as segments of equal length meeting at 120° angles (Figure 23). This method has the disadvantage that two vertices of the cube are represented by the same point.

Figure 21. The typical representation of a cube as two identical squares with edges connected is quite unreal since no cube can ever appear just this way.

Figure 22. Two correct views of a cube are given by the "orthographic" projection (looking straight down) on the left or a foreshortened projection (on the right).

Figure 23. The symmetric "isometric" view of a cube, both solid and transparent, arise by looking at one corner at a 45 degree angle. Then the opposite corner lies directly behind the front corner, so only seven vertices are distinguished in this view.

Figure 24. Two views of a cube in general position: orthographic (on the left) and one-point perspective (on the right).

If we wish a more general image of a cube, we must draw each face as a non-square parallelogram (using a straight-on projection) or as a trapezoid (if we use one-point perspective) (Figure 24). The straight-on (or orthographic) projection is particularly easy to draw since the picture of a cube is completely determined once the position of the edges at one corner is specified. In an orthographic projection, parallel edges of the cube appear as parallel edges in the image, so we can easily complete the picture once we know the position of the three edges at any corner (Figure 25).

Figure 25. In an orthographic drawing, parallel lines in the cube are rendered as parallel lines on the page. Here the full orthographic view of a cube is determined by the orientation of the three edges at any corner.

Once we know how to represent a three-dimensional object on a two-dimensional page or computer graphics screen, we can go on to a much more complicated exercise, that of drawing a four-dimensional analogue of a cube, called a hypercube or tesseract. Many students encounter the idea of a four-dimensional cube in science fiction or fantasy literature, such as Robert Heinlein's story ...and He Built a Crooked House10 or Madeleine L'Engle's A Wrinkle in Time15 or Edwin Abbott Abbott's Flatland.1

Figure 26. By adding a fourth direction to the traditional three-line corner that represents three-dimensional space, we lay a foundation for drawing four-dimensional objects. It shows the direction in which to move a cube to form a four-dimensional hypercube.

Usually a hypercube is constructed by moving an ordinary cube in a direction perpendicular to our space. Although we cannot actually achieve such a motion, we can still draw a picture of what such a construction would look like when the image is projected to a plane (Figure 26). We first draw the cube determined by three of the edges, then move a copy of the cube along the fourth direction and connect corresponding points.

The same procedure enables us to design a three-dimensional model of a four-dimensional cube, using sticks attached by clay balls (as suggested in the last century by Froebel) or more modern materials like drinking straws threaded together with yarn, or some standard building sets. Once again, the full image of a straight-on projection is determined as soon as we specify the four edges coming out of a point (Figure 27).

Figure 27. The completed hypercube formed by connecting corresponding vertices on two copies of a cube.

Just as a foreshortened view of a cube looks like a square within a square with corresponding corners connected, so the analogous foreshortened view of a hypercube looks like a "cube within a cube" with corresponding corners connected (Figure 28).

Figure 28. A foreshortened view of a hypercube, imagined as a cube within a cube with corresponding corners connected.

A cube looks different from different perspectives. A sphere on the other hand always looks like a disc. Any way we look at it, it looks the same. If we mark an equator, then various views give images that are ellipses in different positions (Figure 29). Students also need to be aware of the basic principles of drawing these fundamental forms. It is a fact that a circle always looks like an ellipse, including the extreme case where the ellipse is still a circle or where it degenerates into a doubly covered straight-line segment. Observing this fact makes it easier to draw convincing cylinders and cones (Figure 30).

Figure 29. By rotating a sphere on which an equator has been drawn, it is easy to see that the images of a circle are always some type of ellipse.

Figure 30. To draw cylinders and cones, one begins with an ellipse that represents a suitable perspective as the circular base.

Modern computers are fast enough to produce a sequence of images showing different views of a rotating cube or hypercube, giving the illusion of a three-dimensional object. This process is very familiar to today's students who have grown up with computer-animated special effects and television commercials. We can make use of this experience to give students new appreciation for mathematical forms. As interactive programs become more widely available, students of all ages can have unprecedented opportunities, never before possible, to manipulate and explore geometric forms in three and higher dimensions.

Mathematics Awareness Month is sponsored each year by the Joint Policy Board for Mathematics to recognize the importance of mathematics through written materials and an accompanying poster that highlight mathematical developments and applications in a particular area.