By Hopf H.

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**Extra resources for A New Proof of the Lefschetz Formula on Invariant Points**

**Example text**

To review polygons and nonpolygons, see pages 12–13. vertex—The point of a cone. 34 Cylinder A can is an example of a cylinder. A cylinder is a figure in space that has two parallel bases. The bases are congruent nonpolygons. ← base ← lateral face ← base It also has a curved surface. It has no vertex. The lateral face is the surface that joins the bases. Right Cylinder If the line segment joining the center of the two bases is parallel to the sides and perpendicular to the bases, then the cylinder is called a right cylinder.

Look at the figure below. Then you can create an identical figure to one that exists. This is an important skill to learn in geometry. Look at the figure below. To draw the slide image, first draw a line under the figure and call it L. Place a dot on the line. L Draw the same figure where the dot is located. L This image is called a slide image. It is the same as the original figure, but in a different location on the line. To draw the slide image of the letter B, draw a line under the letter. Place a dot on the line.

42 The figures below are examples of rotations. 1) 2) Flip the figure below and rotate it one turn to the left. original flip rotate Slide the rectangle and then rotate it one turn to the right. original slide rotate A slide moves a figure up, down, or over. A flip produces a mirror image of a figure. A turn rotates a figure around a point. 43 20. Paper Folding Boxes are sold to companies as two-dimensional patterns. Then they are put together to make the shape of the three-dimensional figure. Shipping empty containers as two-dimensional objects saves lots of room on delivery trucks, which saves lots of money.