Show Extra Information Links Hide Extra Information Links Editorial Version

Chapter 3 : Slicing and Contours

Study Questions and Projects

Introduction

• Describe the slices obtained by slicing an orange with a thick skin, or an apple, with or without bumps on the bottom, and with or without a stem. Does it make a difference what way the object is held when the slicing takes place?
• Same question for a stuffed olive or a (seedless) watermelon.
• An object has the "two-piece property" (TPP) if any single straight slice by a long knife cuts it into at most two pieces. Which familiar objects have the TPP and which do not? Consider breakfast table objects, like a fork or a peach or a pear or a banana. What about a cup, a bowl, a glass, or a sugarbowl.
• Does a doughnut or a bagel have the TPP? Give an argument to justify your answer.
• What would other MRI slices look like, in the sagittal, the axial, and the coronal views? What would the "critical slices" look like, where the slice changes form? What would be the difference between looking at MRI images of a living person and MRI images of a skull?
• What would be a good three-dimensional way of describing the lifetime of an oil slick on the ocean?
• Describe the lifetime of a hexagon in Flatland, thought of as an object in ordinary space.
• The lifetime of a woman in Flatland is a two-dimensional object. True or False? Why?
• How could Flatlanders tell the difference between a visit from a sphere and a visit from a double cone coming through point first, parallel to the base? (A double cone is two circular cones stuck together along their bases.)
• What would a visit from a hemisphere look like? What would be the analogous sequence of slices that would occur if we in space were visited by a hyper-hemisphere? (What phenomenon involving a balloon would describe such a visit?)
• Hinton's creatures live on the circular boundary of a disc planet with gravity drawing all objects toward the center. What is being described by the diagram on the top right of page 41? If the eye of one of Hinton's Flatlanders is on the vertical edge, what would each individual see? What are some of the main differences between Hinton's Episode of Flatland and Abbott's Flatland?
• Compare Hinton's Flatland with The Planiverse and Sphereland. Which aspects of An Episode of Flatland are generalized in each of the other books?

Slicing Basic Three-Dimensional Shapes

• In Froebel's display, how are the slices of a cube different when the cube is suspended from the center of an edge or from a vertex?
• How would A Square interpret these slices?
• Where do the slice histories resulting from Froebel's slicing of the cube have critical slices, and what do they look like?
• Describe the various slices of a cylinder, when suspended from the three different eyelets Froebel used.
• Why does the fourth dimension appear to A Square as time?
• If you had a cylindrical glass full of liquid, how could you pour exactly half the liquid into another glass of the same shape?

Slicing from Other Directions

• What would be the slices of a hypercube analogous to those of the cube sliced a plane not parallel to one of its sides?
• Try to imagine the tiling patterns that result when the large cube (made of small cubes) is sliced by a plane which is perpendicular to a long diagonal of the large cube, but which does not go through vertices of the small cubes or through the center of a cube.

Slicing the Hypercube

• What would happen if we sliced a five-dimensional cube starting with a hypercube? With a cube? A square? An edge? A vertex? (To start, think in terms of the sets of vertices.)

Slicing Cylinders

• What are the other slices of the cylinder suspended from its rim?
• What happens when we slice the slices of a four-dimensional cylinder, and how do these relate to the slices of the cylinder in three-space?

Slicing Cones

• What happens when you slice a cylinder by a plane perpendicular to its base?

Contour Lines and Contour Surfaces

• How would the contour surface of Crater Hyperlake and tilted Crater Hyperlake appear?
• Using the contour maps of Twin Peaks and Crater Lake, determine the least steep path a mountain climber could take to reach the various local peaks (maxima). How do the contour lines along these paths compare to the contour lines along other paths?

Slicing Doughnuts and Bagels

• What is required for a surface to be "two-dimensional," in the way that a sphere is two-dimensional?
• Are there any two-dimensional surfaces without "special points," other than the torus?

Originally distributed by Scientific American Library, W. H. Freeman and Co.
All electronic rights reserved, Copyright Thomas F. Banchoff