tag:blogger.com,1999:blog-362699732020-08-14T10:32:53.038-05:00The Mathematical TouristMath Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.comBlogger1715125tag:blogger.com,1999:blog-36269973.post-89065904087061167592020-08-14T07:00:00.037-05:002020-08-14T07:00:02.077-05:00From Point to Hypercube34. Hyperspace HangoutFrom Point to HypercubeBy moving an object in any dimension, you can create an object in the next dimension. Here is how to move from a point to a line segment to a cube to a four-dimensional hypercube.A two-dimensional picture of a three-dimensional model of a four-dimensional hypercube.A point has zero dimensions.Moving a point in a straight line generates a line segment. Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-62330896242416063012020-08-13T07:00:00.009-05:002020-08-13T07:00:01.161-05:00Escaping a Traffic Jam34. Hyperspace HangoutEscaping a Traffic JamImagine that you are traveling on a four-lane highway. You enter a tunnel and get stuck behind a huge, slow truck. You would like to pass the truck, but you are not allowed to change lanes inside the tunnel. You are free to move only in one dimension, so you creep along behind the truck.Now that you've reached the end of the tunnel, you can change lanesMath Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-80195174427889581382020-08-12T07:00:00.002-05:002020-08-12T07:00:04.903-05:00From Video Games to Tic-Tac-Toe34. Hyperspace HangoutFrom Video Games to Tic-Tac-ToeThe cube of your space capsule (see "Hyperspace Hangout") is a weird geometric object in which opposite sides appear connected through a higher dimension. Going through one side brings you back through the opposite side.Many video games have a similar feature in order to keep a figure on the screen. When a game figure moves off the right side Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-49554107280026376322020-08-11T07:00:00.018-05:002020-08-11T07:00:00.286-05:00Hyperspace Hangout34. Hyperspace HangoutYou are zooming randomly from star to star (see "Galactic Gridlock") when suddenly a buzzer sounds, a siren goes off, and the lights in your space capsule begin flashing. Your computer display shows a line in a complicated, knotted pattern.You feel a little scared but excited as you recall Eight's warning. "If your route crosses over and through itself to form a knot, you Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-86225487589228823752020-08-10T07:00:00.035-05:002020-08-11T13:18:40.168-05:00Random Quivers21. Galactic GridlockRandom QuiversIn the early nineteenth century, British botanist Robert Brown traveled around the world collecting plant specimens. He found that certain pollen grains were transparent when examined under a microscope, and he could see distinct particles inside individual grains. Under a microscope these particles appeared to be in continuous motion, zigzagging randomly Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-21007399149230802102020-08-09T07:00:00.024-05:002020-08-11T13:16:51.650-05:00Wandering in Space21. Galactic GridlockWandering in SpaceIn a random walk in three dimensions, a walker has six directions in which to move, chosen randomly, with each step. Instead of flipping a coin or rolling a tetrahedral die (see "Walking a Line" and "Walking on a Grid"), you can use a regular cubic die to decide your direction step by step.Cubic dice have six sides.The resulting paths, however, differ in Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-35962057927931830632020-08-08T07:00:00.037-05:002020-08-11T13:13:21.922-05:00Walking on a Grid21. Galactic GridlockWalking on a GridIt's easy to extend the random walk model to two dimensions. Take steps to the north, south, east, or west, randomly choosing the direction of each step with equal probability. Instead of flipping a coin, you could toss a four-sided (tetrahedral) die.A tetrahedral die has four sides.You can imagine this walk going from vertex to vertex on an infinite Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-60370291659681981222020-08-07T07:00:00.036-05:002020-08-11T13:11:20.290-05:00Walking a Line21. Galactic GridlockWalking a LineImagine yourself balanced on a tightrope high above the ground. You can take steps only forward or backward.Similarly, in a one-dimensional random walk, the walker is confined to a long, narrow path and can step in either of two directions. Tossing a coin would be one way to determine randomly whether to step forward (heads) or backward (tails).You can keep Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-75785537198681902562020-08-06T07:00:00.080-05:002020-08-11T13:09:44.912-05:00Random Walks, Random Knots21. Galactic GridlockRandom Walks, Random KnotsYou're done with your portable music player, so you carefully coil the headphone cord around the player and stuff it in your pocket. The next time you take it out, however, you find that you have to unravel the cord and undo a knot before you can go back to listening to your music.A coiled headphone cord, once unraveled, is likely to contain a Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-55911013910234548042020-08-05T07:00:00.021-05:002020-08-11T13:07:10.953-05:00Galactic Gridlock21. Galactic GridlockIt's daytime when you awaken on the Digits' asteroid (see "The Alien Baseball Field"). Beside you, Anita and Bill are just starting to stir. Most of the Digits are still asleep, but Eight is already up and walking toward you."Good morning! I have a story to tell you," Eight says.Anita, Bill, and you sit down with Eight, facing one another in a square formation."Long, long agoMath Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-89397583199891099882020-08-04T07:00:00.042-05:002020-08-05T13:12:41.402-05:00Spying Pi in the Sky13. Pi in the SkyBack to the Alien DigitsAfter the pie feast (see "Pi in the Sky"), the Digits invite Anita, Bill, and you to spend the night with them camping under the stars.NASASoon the sky grows dark, and the Digits give each of you a sleeping bag. They line up their own sleeping bags in the same order they had sat at the picnic table: 3 1 4 1 5 9 2 6 5…. You and your friends set your Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-90501993748902445632020-08-03T07:00:00.020-05:002020-08-05T13:10:10.661-05:00Targeting Pi13. Pi in the SkyTargeting PiHere's another way to obtain an approximate value of pi.Suppose you have a square dartboard on which you've drawn a circle that just fits within the square. You throw darts so poorly that they land randomly all over the square.Square dartboard with a circle for estimating the value of pi.The circle has a diameter that is exactly the same as the width of the square. Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-59048619026977147032020-08-02T07:00:00.073-05:002020-08-05T13:08:06.439-05:00Digit Hunters13. Pi in the SkyDigit HuntersCalculating the value of pi has been a fascinating challenge since ancient times. In the beginning, knowing pi's value was important for calculating areas, volumes, and other quantities involving land and property. But calculating the value of pi quickly became a pursuit for its own sake. Few practical applications required a value of pi to more than a handful of Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-36494764914791391152020-08-01T07:00:00.034-05:002020-08-05T13:08:39.902-05:00The Digits of Pi13. Pi in the SkyThe Digits of PiThe pie-eating Digits (see "Pi in the Sky") are seated in the same order as the digits of a mysterious and special number in mathematics known as pi.Pi is the ratio of a circle's circumference (the distance around the circle) to its diameter (the distance across, passing through the center). No matter how large or small the circle, the circumference divided by theMath Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-3344566060427616222020-07-31T07:00:00.026-05:002020-08-05T13:03:23.780-05:00Pi in the Sky13. Pi in the SkyAfter a day of baseball and bicycling with the Digit aliens (see "The Alien Baseball Field"), you are growing hungry."Join us for a pie feast!" says Digit Three, leading you to a very long picnic table. Standing at one end of the table, you can't even see all the way to the other end; the table seems to go on forever.The nine Digits scramble off into the woods. Moments later, Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-4889666811915858592020-07-30T07:00:00.068-05:002020-07-31T12:17:53.452-05:00Rolling with Reuleaux8. The Bumpy Bike PathRolling with ReuleauxA circular wheel isn't the only type of wheel that would ride smoothly over a flat, horizontal road.Consider the problem of making a manhole cover with a shape that won't fall through an opening in the street. One answer is to use a circular lid that is slightly larger than the circular hole it covers. The lid can't slip through because it's wider than Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-4294029743341026962020-07-29T07:00:00.035-05:002020-07-30T12:06:28.994-05:00Different Roads for Different Wheels8. The Bumpy Bike PathDifferent Roads for Different WheelsIt turns out that wheels in the shape of other regular polygons, such as pentagons and hexagons, also ride smoothly over bumps made up of pieces of inverted catenaries.The number of sides on the polygon affects the road's shape: as you get more and more sides, the catenary segments required for the road get shorter and flatter. Ultimately,Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-82850588010777083702020-07-28T07:00:00.040-05:002020-07-31T12:25:30.198-05:00Triking Around8. The Bumpy Bike PathTriking AroundStan Wagon, a mathematician at Macalester College in St. Paul, Minnesota, has built a real tricycle with square wheels, which he rides on a road of inverted catenaries.Stan Wagon's square-wheeled trike in action.Wagon first learned about traveling on square wheels when he saw an exhibit at the Exploratorium in San Francisco. The exhibit featured a pair of Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-46967153284049663072020-07-27T07:00:00.023-05:002020-07-30T11:51:56.679-05:00Going Flat8. The Bumpy Bike PathGoing FlatRiding around on a flat tire is no fun. It feels really bumpy. But a square wheel may be the ultimate flat tire. There's no way it can roll over a flat, smooth road without jolting the rider again and again.If the road itself has evenly spaced bumps of just the right shape, however, flat-sided tires can be the secret to a smooth ride!Believe it or not, the bumps onMath Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-59501474292740574962020-07-26T07:00:00.024-05:002020-07-31T12:22:13.169-05:00The Bumpy Bike Path8. The Bumpy Bike PathPuzzler: Why are Treks 6 and 7 missing?Answer: Fibonacci numbers only, please."If you think our baseball field is weird, come and see our roads and tracks," says the tall, thin alien who has been playing first base (see "The Alien Baseball Field").The team of nine Digits leads you, Anita, and Bill down a very bumpy road."How do you digits manage to walk here?" you ask, Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-3687236208093255992020-07-25T07:00:00.032-05:002020-07-26T11:17:49.838-05:00Why a Baseball Could Orbit5. The Alien Baseball FieldWhy a Baseball Could OrbitSuppose you were standing on an extremely high mountaintop on Earth, and you fired a bullet horizontally. The bullet would travel in an arc, curving downward as it speeds away from the mountain and eventually hitting the ground, pulled by gravity.A relatively slow bullet would hit the ground near the mountain. Faster bullets would travel Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-90947284639099385812020-07-24T07:00:00.055-05:002020-07-24T07:00:00.410-05:00Great Circles and Angles5. The Alien Baseball FieldGreat Circles and AnglesThe circular paths formed by the rubber bands around a ball (see "Lines on a Sphere") are called great circles. If you were to slice a ball exactly in half, the rim would be a great circle.On Earth, one geographic example of a great circle would be the equator. The lines of longitude are great circles that intersect one another at the North Pole Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-58886030458220003222020-07-23T07:00:00.029-05:002020-07-26T11:15:52.467-05:00Lines on a Sphere5. The Alien Baseball FieldLines on a SphereAlthough Earth is roughly spherical in shape, its curvature does not affect the geometry of, say, a baseball field (diamond) or a network of city streets, because the planet is so large.NASAYou would have to take Earth's curvature into account, however, if you were plotting an airplane route from Los Angeles to New York and wanted to find the shortest Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-91743346995971726482020-07-22T18:04:00.001-05:002020-07-23T12:02:38.441-05:00Moebius Mentions VFrom a review by Richard Brody of the 1936 Howard Hawks film Come and Get It, in The New Yorker (July 21, 2020):
"It's a mysterious outpost of Hawks's distinctive and original cinematic universe, a tale that seethes with perversity beneath its robust surface; it's a Möbius strip of erotic obsession that anticipates, by more than two decades, Alfred Hitchcock's ultimate sexual doppelgänger drama,Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-74650503058095230442020-07-22T07:00:00.037-05:002020-07-26T11:13:34.855-05:00The Alien Baseball Field5. The Alien Baseball FieldPuzzler: Why is Trek 4 missing?Answer: Four is not a Fibonacci number (see "A Special Sequence").Grasping the computer mouse, you are about to click on the buckyball (see "The Amazing Buckyball"). Oops! The mouse slips, and you inadvertently click on a different object. Your space capsule zooms toward something that looks like a giant baseball."We're landing on a Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0