The Heart of Mathematics: An Invitation to Effective Thinking
The Heart of Mathematics: An Invitation to Effective Thinking
4th Edition
ISBN: 9781118156599
Author: Edward B. Burger, Michael Starbird
Publisher: Wiley, John & Sons, Incorporated
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Concept explainers

Family of Curves
A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These…
XZ Plane
In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are…
Euclidean Geometry
Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) ge…
Lines and Angles
In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.
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Chapter 5.2, Problem 12MS

Möbius lengths. Use the edge identification diagram of a Möhius band to find the lengths of the two bands we get when we cut the Mobius band by hugging the right edge. Give the lengths in terms of the length of the original Möbius band.

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Chapter 5 Solutions

The Heart of Mathematics: An Invitation to Effective Thinking

Ch. 5.1 - Describing distortion. What does it mean to say...Ch. 5.1 - Your last sheet. Youre in your bathroom reading...Ch. 5.1 - Rubber polygons. Find a large rubber band and...Ch. 5.1 - Out, out red spot. Remove the red spot from the...Ch. 5.1 - That theta (S). Does there exist a pair of points...Ch. 5.1 - Your ABCs (H). Consider the following letters made...Ch. 5.1 - Half dollar and a straw. Suppose we drill a hole...Ch. 5.1 - Drop them. Is it possible to take off your...Ch. 5.1 - Coffee and doughnuts (H). Is a standard coffee mug...Ch. 5.1 - Lasting ties. Tie a thin rope around a friends...
Ch. 5.1 - Will you spill? (S). Suppose you rest a glass of...Ch. 5.1 - Grabbing the brass ring. Suppose a string attached...Ch. 5.1 - Hair care. Is a regular comb equivalent by...Ch. 5.1 - Three two-folds. Take three pieces of paper and...Ch. 5.1 - Equivalent objects. Group the objects in this...Ch. 5.1 - Clips. Is a paper clip equivalent to a circle? If...Ch. 5.1 - Pennies plus. Consider the two objects pictured...Ch. 5.1 - Starry-eyed. Consider the two stars below. Are...Ch. 5.1 - Learning the ropes. Pictured below are two ropes,...Ch. 5.1 - HoIy spheres. Consider the two spheres shown. Each...Ch. 5.1 - From sphere to torus. The following sequence of...Ch. 5.1 - Half full, half empty. One glass is half filled...Ch. 5.1 - Male versus female. Consider the male and female...Ch. 5.1 - Holey tori. Are these two objects equivalent by...Ch. 5.1 - More holey tori (H). Are these two objects...Ch. 5.1 - Last holey tori. Are these two objects equivalent...Ch. 5.1 - Beyond the holey inner tube. Suppose you are given...Ch. 5.1 - Heavy metal. Carefully examine this picture of a...Ch. 5.1 - The disk and the inner tube (ExH). Suppose you...Ch. 5.1 - Building a torus (S). Suppose you are given a...Ch. 5.1 - Lasso that hole. Consider the first two tori on...Ch. 5.1 - Knots in dougtnuts. We are given two solid...Ch. 5.1 - From knots to glasses (ExH). Take the thickened...Ch. 5.1 - More Jell-O. Suppose we take a cube of Jell-O,...Ch. 5.1 - Fixed spheres (H). We are given two spheres made...Ch. 5.1 - Holes. Is a torus equivalent to a two-holed torus?...Ch. 5.1 - More holes. Is a two-holed torus equivalent to a...Ch. 5.1 - Here we celebrate the power of algebra as a...Ch. 5.1 - Here we celebrate the power of algebra as a...Ch. 5.1 - Here we celebrate the power of algebra as a...Ch. 5.1 - Here we celebrate the power of algebra as a...Ch. 5.1 - Here we celebrate the power of algebra as a...Ch. 5.2 - One side to every story. What is a Mobius band?Ch. 5.2 - Maybe Mobius. How can you look at a loop of paper...Ch. 5.2 - Singin the blues. Take an ordinary strip of white...Ch. 5.2 - Whos blue now? Take an ordinary strip of white...Ch. 5.2 - Twisted sister. Your sister holds a strip of...Ch. 5.2 - Two twists. Take a strip of paper, put two half...Ch. 5.2 - Two twists again. Take a strip of paper, put two...Ch. 5.2 - Three twists (H). Take a strip of paper, put three...Ch. 5.2 - Prob. 11MSCh. 5.2 - Möbius lengths. Use the edge identification...Ch. 5.2 - Squash and cut. Take a Möbius band and squash it...Ch. 5.2 - Two at once. Take two strips of paper and put them...Ch. 5.2 - Parallel Möbius. Is it possible to have two...Ch. 5.2 - Puzzling. Suppose you have a collection of jigsaw...Ch. 5.2 - Möbius triangle. Make a 1-inch-wide Möbius band,...Ch. 5.2 - Thickened Möbius. Imagine a Möbius band...Ch. 5.2 - Thickened faces. How many faces (sides) does a...Ch. 5.2 - Thick then thin. Suppose we take a Môbius band,...Ch. 5.2 - Drawing the band (ExH). Imagine you have a Möbius...Ch. 5.2 - Tubing (H). Suppose we take two Möbius bands and...Ch. 5.2 - Bug out (ExH). Suppose you are a ladybug on the...Ch. 5.2 - Open cider. Consider the Klein bottle half filled...Ch. 5.2 - Rubber Klein (S). Suppose you have a rectangular...Ch. 5.2 - One edge. Using the method on page 347 for...Ch. 5.2 - Twist of fate (S). Using the edge-identification...Ch. 5.2 - Linked together. Using the edge-identification...Ch. 5.2 - Count twists. Using the edge-identification...Ch. 5.2 - Dont cross. Can you draw a curve that does not...Ch. 5.2 - Twisted up (H). Suppose you are given a band of...Ch. 5.2 - Prob. 32MSCh. 5.2 - Find a band. Find a Möbius band on the surface of...Ch. 5.2 - Holy Klein. Show that the figure on the left is...Ch. 5.2 - Möbius Möbius. Show that the Klein bottle is two...Ch. 5.2 - Attaching tubes. Consider a Möbius band with two...Ch. 5.2 - Möbius map (H). Using felt-tip color pens that...Ch. 5.2 - Thick slices. Thicken a Môbius band and then...Ch. 5.2 - Bagel slices. If we take a bagel and slice it in...Ch. 5.2 - Gluing and cutting. Consider a rectangular sheet...Ch. 5.2 - Here we celebrate the power of algebra as a...Ch. 5.2 - Here we celebrate the power of algebra as a...Ch. 5.2 - Here we celebrate the power of algebra as a...Ch. 5.2 - Here we celebrate the power of algebra as a...Ch. 5.2 - Here we celebrate the power of algebra as a...Ch. 5.3 - Knotty start. Which of the followign knots are...Ch. 5.3 - The not knot. What is the unknot?Ch. 5.3 - Crossing count. Count the crossings in each knot...Ch. 5.3 - Tangled up. Is the figure below a knot or a link?Ch. 5.3 - Ringing endorsement. What are the Borromean rings?Ch. 5.3 - Human trefoil. What is the minimum number of...Ch. 5.3 - Human figure eight. What is the minimum number of...Ch. 5.3 - Stick number (ExH). What is the smallest number...Ch. 5.3 - More Möbius. Make a Möbius band with three half...Ch. 5.3 - Slinky (H). Take a Slinky, lengthen one of its...Ch. 5.3 - More slink. Take a Slinky, and this time weave an...Ch. 5.3 - Make it. Use a piece of string or an extenstion...Ch. 5.3 - Knotted (S). Take an unknotted loop. Tie a knot in...Ch. 5.3 - Slip. Take an unknotted loop and put a slip knot...Ch. 5.3 - Dollar link. Take two paper clips and a dollar and...Ch. 5.3 - Prob. 18MSCh. 5.3 - Unknotting knots (H). In each of the two knots at...Ch. 5.3 - Alternating. A picture of a knot is alternating...Ch. 5.3 - Making it alternating. Consider the knot on the...Ch. 5.3 - Prob. 22MSCh. 5.3 - One cross (H). Prove that any loop with exactly...Ch. 5.3 - Two loops (S). Is there a picture of two linked...Ch. 5.3 - Hold the phone. Disconnect the wire from the phone...Ch. 5.3 - More unknotting knots. In these two knots, find...Ch. 5.3 - Unknotting pictures (S). Suppose you are given a...Ch. 5.3 - Twisted. Suppose we are given a figure consisting...Ch. 5.3 - More alternating. First reread Mindscape 20. For...Ch. 5.3 - Crossing numbers. Suppose you are given pictures...Ch. 5.3 - Lots of crossings. Suppose you arc given a picture...Ch. 5.3 - Torus knots (H). Can you draw a trefoil knot on a...Ch. 5.3 - Two crosses. Prove that any loop with exactly two...Ch. 5.3 - Hoop it up. Show that every knot can be positioned...Ch. 5.3 - The switcheroo. Pictured below is a way of...Ch. 5.3 - 4D washout. Why is the study of knots and links...Ch. 5.3 - Brunnian links (H). Link four loops together in...Ch. 5.3 - Fire drill (ExH). A fire starts in your...Ch. 5.3 - Fixed spheres again. We are given two spheres that...Ch. 5.3 - Here we celebrate the power of algebra as a...Ch. 5.3 - Here we celebrate the power of algebra as a...Ch. 5.3 - Here we celebrate the power of algebra as a...Ch. 5.3 - Here we celebrate the power of algebra as a...Ch. 5.3 - Here we celebrate the power of algebra as a...Ch. 5.4 - Fixed things first. What does the Brouwer Fixed...Ch. 5.4 - Say cheese. Youre making an open-faced cheese...Ch. 5.4 - Fixed flapjacks. Youre making pancakes and...Ch. 5.4 - Prob. 4MSCh. 5.4 - Loop around. What does the Hot Loop Theorem...Ch. 5.4 - Fixed on a square. Does the Brouwer Fixed Point...Ch. 5.4 - Fixed on a circle. Does the Brouwer Fixed Point...Ch. 5.4 - Winding arrows. In each drawing below we have a...Ch. 5.4 - Prob. 10MSCh. 5.4 - Prob. 11MSCh. 5.4 - Home heating (H). Prove that there are two points...Ch. 5.4 - Prob. 13MSCh. 5.4 - Prob. 14MSCh. 5.4 - Prob. 15MSCh. 5.4 - Lining up (H). Suppose we have two line segments...Ch. 5.4 - A nice temp. Must there be two antipodal points on...Ch. 5.4 - Prob. 18MSCh. 5.4 - Diet drill. Suppose someone weighs 160 lbs. and...Ch. 5.4 - Speedy (S). You enter a tollway and are given a...Ch. 5.4 - The cut core. Suppose we have the red and blue...Ch. 5.4 - Fixed without boundary. Do you think that the...Ch. 5.4 - Take a hike (ExH). A hiker decides to climb up...Ch. 5.4 - Here we celebrate the power of algebra as a...Ch. 5.4 - Here we celebrate the power of algebra as a...Ch. 5.4 - Here we celebrate the power of algebra as a...Ch. 5.4 - Here we celebrate the power of algebra as a...Ch. 5.4 - Here we celebrate the power of algebra as a...

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