Exercises in imagining

How do you imagine geometric figures in your head? Most people talk about their three-dimensional imagination as `visualization', but that isn't exactly right. A visual image is a kind of picture, and it is really two-dimensional. The image you form in your head is more conceptual than a picture-you locate things in more of a three-dimensional model than in a picture. In fact, it is quite hard to go from a mental image to a two-dimensional visual picture. Children struggle long and hard to learn to draw because of the real conceptual difficulty of translating three-dimensional mental images into two-dimensional images.

Three-dimensional mental images are connected with your visual sense, but they are also connected with your sense of place and motion. In forming an image, it often helps to imagine moving around it, or tracing it out with your hands. The size of an image is important. Imagine a little half-inch sugarcube in your hand, a two-foot cubical box, and a ten-foot cubical room that you're inside. Logically, the three cubes have the same information, but people often find it easier to manipulate the larger image that they can move around in.

Geometric imagery is not just something that you are either born with or you are not. Like any other skill, it develops with practice.

Below are some images to practice with. Some are two-dimensional, some are three-dimensional. Some are easy, some are hard, but not necessarily in numerical order. Find another person to work with in going through these images. Evoke the images by talking about them, not by drawing them. It will probably help to close your eyes, although sometimes gestures and drawings in the air will help. Skip around to try to find exercises that are the right level for you.

When you have gone through these images and are hungry for more, make some up yourself.

1. Picture your first name, and read off the letters backwards. If you can't see your whole name at once, do it by groups of three letters. Try the same for your partner's name, and for a few other words. Make sure to do it by sight, not by sound.

2. Cut off each corner of a square, as far as the midpoints of the edges. What shape is left over? How can you re-assemble the four corners to make another square?

3. Mark the sides of an equilateral triangle into thirds. Cut off each corner of the triangle, as far as the marks. What do you get?

4. Take two squares. Place the second square centered over the first square but at a forty-five degree angle. What is the intersection of the two squares?

5. Mark the sides of a square into thirds, and cut off each of its corners back to the marks. What does it look like?

6. How many edges does a cube have?

7. Take a wire frame which forms the edges of a cube. Trace out a closed path which goes exactly once through each corner.

8. Take a rectangular array of dots in the plane, and connect the dots vertically and horizontally. How many squares are enclosed?

9. Find a closed path along the edges of the diagram above which visits each vertex exactly once? Can you do it for a array of dots?

10. How many different colors are required to color the faces of a cube so that no two adjacent faces have the same color?

11. A tetrahedron is a pyramid with a triangular base. How many faces does it have? How many edges? How many vertices?

12. Rest a tetrahedron on its base, and cut it halfway up. What shape is the smaller piece? What shapes are the faces of the larger pieces?

13. Rest a tetrahedron so that it is balanced on one edge, and slice it horizontally halfway between its lowest edge and its highest edge. What shape is the slice?

14. Cut off the corners of an equilateral triangle as far as the midpoints of its edges. What is left over?

15. Cut off the corners of a tetrahedron as far as the midpoints of the edges. What shape is left over?

16. You see the silhouette of a cube, viewed from the corner. What does it look like?

17. How many colors are required to color the faces of an octahedron so that faces which share an edge have different colors?

18. Imagine a wire is shaped to go up one inch, right one inch, back one inch, up one inch, right one inch, back one inch, .... What does it look like, viewed from different perspectives?

19. The game of tetris has pieces whose shapes are all the possible ways that four squares can be glued together along edges. Left-handed and right-handed forms are distinguished. What are the shapes, and how many are there?

20. Someone is designing a three-dimensional tetris, and wants to use all possible shapes formed by gluing four cubes together. What are the shapes, and how many are there?

21. An octahedron is the shape formed by gluing together equilateral triangles four to a vertex. Balance it on a corner, and slice it halfway up. What shape is the slice?

22. Rest an octahedron on a face, so that another face is on top. Slice it halfway up. What shape is the slice?

23. Take a array of dots in space, and connect them by edges up-and-down, left-and-right, and forward-and-back. Can you find a closed path which visits every dot but one exactly once? Every dot?

24. Do the same for a array of dots, finding a closed path that visits every dot exactly once.

25. What three-dimensional solid has circular profile viewed from above, a square profile viewed from the front, and a triangular profile viewed from the side? Do these three profiles determine the three-dimensional shape?

26. Find a path through edges of the dodecahedron which visits each vertex exactly once.