## Exploring Space and Shape

Problems and tasks from a variety of sources intended to illustrate the way mathematics arises in life and work.

Patterns. A square tile is inscribed with four quarter-circles, each centered at a corner of the square and touching at the midpoints of the sides of the square. Inside the four-pointed star described by these circles is inscribed another square, and inside this another star, etc. What is the ratio of the area of one square to the next in this series?

Inscribed Triangles. Determine, for a given triangle ABC, the area of an inscribed triangle whose vertices divide the three sides of ABC in the same proportion.

Planimeters. What is a planimeter and how does it work? Why is it used? Are any still used today? Either build a working model, or explain in words and diagrams how to build one.

Crossed Triangles. Two triangles ABC and ADE share a common angle at vertex A. AEB are collinear, with AE = 2 cm and EB = 3 cm. ACD are collinear, with AC = 3. If P is the intersection of BC and DE, and if BP = 4 cm, DP = PE = 2 cm, find the length of DC.

Shortest Chord. Suppose P is a point in the plane and m and n are two intersecting lines in the same plane. What is the shortest line segment (chord) through P that touches m and n?

Shortest Paths. Exploring shortest paths in cities with skyways (or subways) connecting buildings leads to three dimensional "taxi-cab" geometry problems. What is the shortest distance between two points in such a city? What is the shortest distance between two points on the surface of a square column? On the surface of a cube?

Expanding Pyramids. Imagine a right pyramid with a rectangular base. Suppose that each side of the pyramid is translated outward by a fixed amount without changing its size or shape. Describe the shape of the object that is framed by the expanded sides.

Polygons in 3-Space. For which N are there polygons in 3- space with N equal sides and only right angles?

Common Tangents. How many common tangent lines might two circles have?

Curves. Given a piece of bent coat hanger, describe how one could determine whether the shape is circular, or whether it matches some part of a parabola.

Cuts and Slices. Divide the Euclidean plane with a random line, then do it again, and again, ... . Into how many regions do these cuts divide the plane? What happens if one repeats this exercise on a circular disk rather than the entire plane? Do the same patterns appear?