July 31, 2020

Pi in the Sky

13. Pi in the Sky

After 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, Digit Three reappears with a humongous pizza and sits down in a special chair at the end of the table. Then Digit One appears with a meat pie and sits on the long bench to the left of Three. Four comes along with a spinach pie and sits beside One.

More Digits emerge from the woods, each carrying a different kind of pie. Another One sits beside Four with a cherry pie. Then comes Five with a blueberry pie, Nine with a peach pie, and Two with an apple pie.

As Six arrives to take a seat beside Two, the Digits invite you, Anita, and Bill to sit across the table. More digits keep coming, and they start passing around slices of pie to share.

"How many of you are coming?" you ask.

"Everyone is invited, and our current population is about 50 trillion," says Three, who is still sitting at the head of the table. "Our population keeps growing because we never die, and we keep on adding Digits."

As you contemplate the idea of eating pie with more than 50 trillion Digits, Bill looks up and down the row of Digits on the bench. He gets a befuddled look on his face. Anita studies the row of Digits and breaks out laughing.

What is so funny about the seating order?

July 30, 2020

Rolling with Reuleaux

8. The Bumpy Bike Path

Rolling with Reuleaux

A 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 the hole, no matter which way you turn it. A circle has the same width no matter where you start on the edge to cross through the middle.

A standard circular manhole cover.

In contrast, an oval (or ellipse) is longer than it is wide. You can always find a way to slip an oval lid through an oval hole that is the same size or slightly smaller. That's also true of a square cover or a six-sided, or hexagonal, cover.

Amazingly, the circle isn't the only shape that would work safely as a manhole cover. Would that shape also make a smooth-riding wheel on a flat surface?

One possibility is the Reuleaux triangle, named after engineer Franz Reuleaux, who was a teacher in Berlin, Germany, more than a hundred years ago.

The Reuleaux triangle, unlike a standard equilateral triangle (dashed lines), has curved sides.

You might find an example of a Reuleaux triangle in your medicine cabinet. If you turn a bottle of NyQuil cough medicine or Pepto-Bismol stomach medicine upside down, the shape you see looks like a Reuleaux triangle. If you try rolling one of these bottles on its side, you'll find that it rolls nearly as smoothly as a round bottle.

The bottom of a bottle of NyQuil cough medicine has the shape of a triangle with curved sides, somewhat similar to a Reuleaux triangle.

One way to draw a Reuleaux triangle is to start with an equilateral triangle, which has three sides of equal length. Place the pointed end of a pair of compasses at one corner of the triangle and stretch the arms until the pencil reaches another corner. Then draw an arc between two corners of the triangle. Draw two more arcs centered on the triangle's other corners.

Reuleaux's "curved triangle," as he called it, has a constant width—just like a circle. It can roll smoothly on a flat surface, like a circular wheel.

A pair of wheels shaped like Reuleaux triangles rolling along track in a display at the Exploratorium in San Francisco.

In fact, you can make a manhole cover or a wheel out of any regular polygon with an odd number of sides. Beginning with a five-sided shape (regular pentagon), for example, you can construct a rounded pentagonal shape with a constant width.

Reuleaux polygons with curved sides based on a regular pentagon (left) and regular heptagon (right).

Any wheel or other object with such a cross section would roll smoothly across your kitchen floor or down the street.

A manhole cover shaped like a Reuleaux triangle, found in San Francisco.

Imagine walking down the street and finding differently shaped manhole covers on every block!

July 29, 2020

Different Roads for Different Wheels

8. The Bumpy Bike Path

Different Roads for Different Wheels

It 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, the wheel has so many sides that it looks like a  circle and its road is practically flat.

Triangular wheels don't work, though. As a triangle rolls over one catenary, it ends up bumping into the next catenary.

A triangular wheel doesn't work because its points get stuck in the roadbed.

Mathematicians have found roads for other wheel shapes, including an ellipse (which looks like a flattened circle), a cardiod (like a heart with a rounded tip), a four-lobed rosette (like a flower with four petals), and a teardrop.

Roadways for a four-lobed rosette (above) and a teardrop shape (below).

You can even start with a road profile and find the wheel shape that runs smoothly across it. A sawtooth road, for example, requires a wheel pasted together from pieces of an equiangular spiral (sort of a cross between a flower and a star shape).

So far, no one has ever found a road-wheel combination in which the road has the same shape as the wheel. That's an intriguing challenge for mathematicians.

Various combinations of wheel shape and road profile would give you a smooth ride.

July 28, 2020

Triking Around

8. The Bumpy Bike Path

Triking Around

Stan 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 square wheels joined by an axle riding over an inverted catenary roadbed.

Square wheels on display at the Exploratorium in San Francisco.

Intrigued by the demonstration, Wagon decided to build a square-wheeled bike he could actually ride himself. "As soon as I saw it could be done, I had to do it," he said.

The bike was tricky to build, and Wagon ended up with an unusual kind of tricycle. He also had to construct an inverted catenary surface on which to ride it. Wagon's square-wheeled trike went on display at the Macalester College science center.

"It rides like a normal bike, though steering is difficult," said Wagon. If you turn the square wheels too much, they get out of sync with the inverted catenaries.

Having one front wheel and two back wheels helps make it easier to ride the trike in a straight line. If you had a bicycle instead, you would need to turn the wheel now and then to help keep your balance, and that's hard to do with square wheels on a bumpy road.

For the trike-roadway combination to work, the length of the side of a square wheel also has to match the length of one unit of an inverted catenary road. So if you start with one corner of the square at the beginning of the road, each corner will end up at a point between two catenaries as the wheel rolls along.

At the National Museum of Mathematics in New York City, trikes with square wheels ride on a circular pathway of inverted catenaries.

July 27, 2020

Going Flat

8. The Bumpy Bike Path

Going Flat

Riding 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 on the aliens' road (see "The Bumpy Bike Path") are the perfect shape to produce a smooth ride on a square-wheeled bike. It can't be just any old square-wheeled bike, though. Each side of the square tires must be related mathematically to the bump's height.

The bumps on the aliens' road are the shape of upside-down, or inverted, catenaries. A catenary is the curve formed by a chain or rope hanging loosely between two supports.

A hanging chain, fastened at its two ends, forms a catenary shape.

Create your own inverted catenary.

You will need:
  • piece of string, rope, or chain about 10 inches (25 centimeters) long
  • sheet of paper
What to do:
  1. Hold one end of the string in your left hand and the other end in your right hand, each hand at about the same height.
  2. Let the string hang loosely between your hands. It forms a catenary!
  3. Carefully lay it down on a sheet of paper without changing the string's shape.
  4. Turn the paper so the string forms an upside-down U.

Now you can picture the cross section of each bump of the alien roadway. A series of such bumps, lined up in a row, would be the perfect surface on which to ride a square-wheeled bike.

How a square wheel rolls over a roadbed consisting of a sequence of inverted  catenaries. Wikipedia

July 26, 2020

The Bumpy Bike Path

8. The Bumpy Bike Path

Puzzler: 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, trying to keep your balance.

"We don't often walk," says Digit One, the tall, thin alien, with a wide grin. "We ride our bikes."

"Yeah, sure," says Anita."Who could ride a bike on these bumps?" 

"Take a look at our bikes," says Digit One, pointing to a nearby hut with a sign.

Digit Four opens the door, revealing several odd-looking bikes. "Why don't you try them out," the alien tells you and your two companions. "You might be surprised how smooth the ride can be if your bike suits the road."

The first bike you see looks ordinary, but the one beside it has wheels that look like flattened circles, or ellipses. A third bike has square wheels. There's even a bike with star-shaped wheels.

Four types of bike wheels: round (top left), elliptical (top right), square (bottom left), and star-shaped (bottom right).

Which of these four bikes will give you the smoothest ride on the bumpy road?

July 25, 2020

Why a Baseball Could Orbit

5. The Alien Baseball Field

Why a Baseball Could Orbit

Suppose 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 farther. If it were fast enough, a bullet could end up going entirely around Earth. Like a tiny moon or a satellite, it would travel around in its circular orbit again and again. (Be sure to duck before it circles the globe and comes back to you from behind.)

A bullet shot horizontally from a mountaintop will eventually hit the ground, unless its initial speed is fast enough to put it in orbit.

Using the laws of motion formulated by the English scientist Isaac Newton more than three hundred years ago, it's possible to calculate the necessary speed for an object, such as a bullet or a spacecraft, to go into orbit around Earth or any other spherical body.

The launch speed (escape velocity) depends on the object's distance from the sphere's center and the acceleration caused by gravity. A satellite orbiting just above Earth's surface would have to travel at a speed of about 11.2 kilometers per second to remain in orbit.

A softly thrown baseball moves at about 10 meters per second. A hard-hit baseball can go 40 meters per second or faster. That's fast enough for a ball to go into orbit around a small asteroid!

July 24, 2020

Great Circles and Angles

5. The Alien Baseball Field

Great Circles and Angles

The 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 and at the South Pole.

Great circles are the largest of all circles that can be drawn on the surface of a sphere.

These great circles divide up the surface of a sphere into various regions, such as the curved triangle shown.

The shortest distance between two points on a sphere is along the arc of a great circle joining the two points. On any three-dimensional surface, including a sphere, the shortest distance between two points is called a geodesic.

To find the geodesic between any two points marked on a baseball or a tennis ball, just stretch a rubber band around the ball to form a great circle that passes over both points.

A baseball diamond on a small, spherical asteroid would have baselines that are arcs of great circles. That's why the baselines end up being curved (see "Alien Baseball Field").

Four great circles intersect to form a "square."

Curved baselines are not the only weird things about an asteroid baseball diamond. If you were to measure the angles at the four corners of the diamond, you would get a surprising result.

On a flat surface, the angles are each 90 degrees, but on a spherical surface, they are not. Furthermore, the sum of the four angles is 360 degrees on a flat baseball diamond, but not on a sphere. How large are the angles?

The baseball diamond is a square shape, with all four angles the same size. On a sphere, each angle of a square is greater than 90 degrees. So, the sum of the angles of a four-sided figure on a sphere is greater than 360 degrees. The larger the diamond (square), the larger the angles on any given sphere.

The sum of the angles of a triangle on a sphere is greater than 180 degrees.

Great Soccer Circles

Study the edges of the pentagons and hexagons on a classic soccer ball. Try covering the edges with large rubber bands. Are all of the "edge" lines on a soccer ball arcs of great circles?

Pentagons and hexagons happen to fit together so their edges all meet on great arcs on a sphere. Because these shapes are rounded on a sphere, their angles are larger than the angles of regular pentagons and hexagons that are flat.


A pilot flies due south 100 kilometers, then goes east 100 kilometers, then north 100 kilometers. She ends up right back where she started. Where was her starting point? (Hint: Study a world globe.)

Answer: The simplest answer is that the pilot started at the North Pole. It is also possible, however, that the pilot started somewhere on a great circle that is 116 kilometers from the South Pole. After flying south 100 kilometers, then east 100 kilometers, she would have completed a circle around the South Pole. Then, when she goes north 100 kilometers, she ends up right back where she started.

See also "Touring the Poles."


Pilots and airline route planners covering large distances across the globe have to consider Earth's shape in their calculations. The shortest route from New York to Tokyo, for example, doesn't go directly from east to west along a line of latitude, but actually follows a great circle that passes near the North Pole. Drawn on a flat map, such a route may look curved, but it's really a shortcut for both pilot and passengers.

In fact, spherical geometry plays an important role not just in navigation but also in many other fields. In mathematics it appears in trigonometry, topology, calculus, and other areas.

Spherical geometry also has applications in physics, chemistry, crystallography, earth sciences, astronomy, art, technical drawing, industrial design, and engineering. You couldn't put a satellite into orbit around Earth or send a spacecraft to Mars without understanding spherical geometry.

NEXT: Why a Baseball Could Orbit

July 23, 2020

Lines on a Sphere

5. The Alien Baseball Field

Lines on a Sphere

Although 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.


You 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 possible path. The standard rules of geometry on a flat surface would no longer apply.

On a small, rounded asteroid, the asteroid's curvature could even affect the shape of a baseball diamond—if the asteroid were small enough.

On a flat surface, the shortest path between two points is a straight line. What is the shortest path between two points on a spherical surface?

Explore a spherical surface to find the shortest distance between two points.

You will need:
  • baseball or tennis ball
  • three rubber bands
What to do:
  1. Stretch a rubber band around the ball so it makes as large a loop, or ring, as possible.
  2. Stretch a second rubber band around the ball at a different angle, again making as large a loop as possible. The two rubber bands should intersect at two different points on the ball. The two points where they intersect should be opposite each other, like the North and South Poles on Earth.
  3. Stretch the third rubber band around the ball, again making as wide a ring as possible, and see where it intersects the first two rings. Experiment with putting the third band in different positions to see how it can intersect the first two rings in different ways.
  4. Pick any two points where the rubber bands intersect. The shortest distance between them on the ball's surface is along the rubber band connecting the two points.

How to place the rubber bands around the ball.

July 22, 2020

Moebius Mentions V

From 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,'Vertigo.'"

See also "Moebius Mentions I, II, III, IV."

The Alien Baseball Field

5. The Alien Baseball Field

Puzzler: 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 baseball asteroid!" Bill exclaims.

"What's that?" asks Anita, pointing out the window at some markings on the ground. "It's like a baseball diamond!"

"What a weird playing field," Bill says. "The baselines look sorta curved."

As the three of you step outside, a tall, very thin figure approaches. "Hey, you're late for the game," the alien says.

You gaze in awe at this odd figure, which looks amazingly like the numeral "1."

"Come on," the figure says, leading you to the curvy baseball diamond. "It's Digits versus Earthlings; you guys bat first. We can at least get started."

As the tall, thin Digit walks over to first base, you spot another strange figure standing near second base: an alien creature shaped like the number "2." What's more, there's a "3" standing at third base, a "4" behind home plate, a "5" at shortstop, and a "6" on the pitcher's mound.

The outfield seems to disappear beyond the horizon, but you can make out the top of a "7" in left field. In center field, you see the top of an "8," and the right fielder looks like a "9."

"You first," says Anita, picking up a bat and handing it to you.

The pitcher sends you a slow, easy pitch, and you give the ball a light tap. It's a grounder heading straight toward third. Before you can run, however, the ball curves and ends up rolling across the baseline. What an unfair foul!

When the next pitch comes, you swing and smack the ball toward center field. Instead of landing in the outfield, though, the ball keeps on going until it vanishes over the horizon. You race around the bases, wondering why you don't have to turn sharply at each corner.

As you approach home plate, the ball you had hit reappears from behind the backstop, just misses the catcher, and flies over the field a second time.

What sort of shape is the baseball diamond, and why doesn't the baseball land?

July 21, 2020

Engineer from "Spaceship Earth"

3. The Buckyball Asteroid

Engineer from "Spaceship Earth"

R. Buckminster Fuller, known familiarly as Bucky, lived from 1895 to 1983. He grew up and lived to witness the invention of the automobile, airplane, radio, television, electronic computer, and atomic bomb.

The Wright Flyer, on display at the National Air and Space Museum in Washington, D.C., became the first successful heavier-than-air powered aircraft in 1903.

At a time when many people regarded technology as a means for profit and for waging war, Bucky believed that we could use it to help abolish poverty, hunger, and conflict. He called our planet "Spaceship Earth" to help convince people that we all need to work together as the crew of a ship does.

Bucky had an original way of thinking that made it hard for him to fit into the status quo. He was twice expelled from Harvard College.

The first time, when he was expelled for excessive partying, his family sent him off to a Canadian cotton mill as an apprentice machinist, hoping he would become more mature and responsible. At the mill, he became so good at making, installing, and troubleshooting complex machinery that Harvard invited him back.

The second time, he was expelled for "showing insufficient interest in his studies." Bucky went to work in a meat-packing house and never completed college.

During World War I, he served as a radio operator on a ship with wireless communications. Using his mechanical skills and ingenuity, he helped design a system for rescuing pilots who had been shot down over water.

The navy rewarded him with an opportunity to study briefly at the U.S. Naval Academy in Annapolis, where he enjoyed courses that went beyond academic theory and dealt with the realities of global communications, air travel, and logistics. The skills he developed in his various mechanical ventures would become very important to his future work as an engineer, inventor, and architect.

One of Bucky's later projects was to examine a system of geometry based on the tetrahedron. That led to the novel building design that made him famous: the geodesic dome.

Geometrically, a geodesic dome is akin to a sphere with a piece sliced off its bottom. Like an icosahedron, a classic geodesic sphere has twenty triangular sides. What makes a geodesic dome different from an icosahedron is the fact that its triangles are slightly curved and each triangle is subdivided into smaller triangles. The corners of all these smaller triangles are each the same distance from the sphere's center.

In a geodesic sphere, each face of a polyhedron is subdivided into triangles.

Unlike conventional buildings, Fuller's geodesic domes become stronger, lighter, and cheaper per unit of volume as their size increases. They enclose the largest possible volume of space using the smallest possible surface area.

Since Bucky patented his design in 1947, hundreds of thousands of geodesic domes have been built around the world, everywhere from high mountaintops to the South Pole.

The U.S. Pavilion (above) at Expo 67 in Montreal was a two-hundred-foot-high version of Buckminster Fuller's geodesic dome. The close-up (below) shows the hexagonal units that make up the surface. The dome is now a family-oriented museum called the Biosphere.

These domes are among the strongest structures for their weight ever devised and least vulnerable to damage from hurricanes and earthquakes.

July 20, 2020

Buckyballs Everywhere

3. The Buckyball Asteroid

Buckyballs Everywhere

Interest in the truncated icosahedron (buckyball as a molecule) dates back to Archimedes, a Greek mathematician and inventor who lived in the third century B.C. Others probably conceived of the truncated icosahedron even earlier. In fact, objects based on this shape show up in many cultures throughout the world.

Sepak raga, a popular game in Southeast Asia, uses a woven ball that looks similar to a soccer ball.

Woven ball used for the game sepak raga.

In Mozambique, Zaire, Brazil, and many other countries, local artists weave baskets in patterns with regular hexagonal holes. To turn a flat hexagonal weave into a curved basket shape, the weaver performs a mathematical trick: By reducing the number of strands, they make certain holes pentagonal instead of hexagonal.

July 19, 2020

The Amazing Buckyball

3. The Buckyball Asteroid

The Amazing Buckyball

One of the shapes you saw on the space-capsule screen (see "The Buckyball Asteroid") is a truncated icosahedron.

Truncated icosahedron.

A regular icosahedron is made up of twenty equilateral triangles that meet at twelve points, or vertices. To create a truncated icosahedron, you chop off each vertex so the twelve vertices turn into twelve regular pentagons and the twenty equilateral triangles become twenty regular hexagons.

Lopping off one corner of an icosahedron (above) is the first step in creating a truncated icosahedron (below).

The resulting shape has twenty hexagons and twelve pentagons as its surface, thirty-two faces in all.

Pattern showing all thirty-two faces for constructing your own truncated icosahedron (buckyball).

Three-dimensional geometry is very useful for describing the arrangement of atoms in different materials. Carbon atoms, for example, can arrange themselves in a pattern that looks a lot like neatly stacked tetrahedra. Arranged in this way, carbon atoms form diamonds, one of the hardest materials known.

Tetrahedral arrangement of carbon atoms in diamond.

In contrast, when carbon atoms are arranged in hexagonal rings linked together into to form vast sheets, they form graphite, a soft material used in lubricants and in pencils. The hexagonal pattern looks like a honeycomb grid (see "Paving the Plane").

In the 1980s, scientists discovered carbon molecules in the shape of truncated icosahedra. Each molecule consists of sixty carbon atoms. They named the molecule buckminsterfullerene, after R. Buckminster Fuller, the engineer, mathematician, and architect who had studied and designed buildings with a similar structure. Scientists sometimes call these molecules "buckyballs" for short.

Computer-generated model of a buckyball molecule, which consists of sixty carbon atoms arranged to form a spherical cage.

If you count the number of hexagons and pentagons on the surface of a classic soccer ball, you will realize that when you play soccer, you are kicking a truncated 
icosahedron, or buckyball!

Classic soccer ball.

Actually, it's not quite a truncated icosahedron, because the pentagons and hexagons on a true mathematical buckyball are flat. On a soccer ball, they are rounded so that the ball is essentially spherical.

Show that Euler's Rule (see "Solid Faces") for the number of faces, vertices, and edges on a polyhedron works for a truncated icosahedron (buckyball).

A soccer ball has 32 faces (20 hexagons and 12 pentagons), so F = 32.
A soccer ball is akin to a truncated icosahedron, so its 12 pentagons come from "slicing off" the 12 vertices of an icosahedron. Therefore, a truncated icosahedron has 12 ✕ 5 = 60 vertices.
F + V − E = 32 + 60 −  E = 2. Solving the equation gives 90 as the number of edges.

July 18, 2020

Solid Faces

3. The Buckyball Asteroid

Solid Faces

The shapes pictured on your space capsule's screen are examples of geometric figures known as solids (see "The Buckyball Asteroid"). Solids have three dimensions: length, width, and height. Many solid objects, from pyramids and dice to baseballs and cereal boxes, have shapes that can be described in simple geometric terms.

Dice serve as examples of the solid geometric shape known as a cube.

A solid formed by polygons that enclose a single region of space is called a polyhedron. Five of the six shapes you saw on the screen had surfaces made up identical, regular polygons, which meet at each corner, or vertex, in exactly the same way.

A regular tetrahedron has four faces, each one an equilateral triangle. Here's what a tetrahedron might look like if it were cut open and unfolded into a flat shape.

A regular tetrahedron, as seen whole (left) and cut open and unfolded (right), has four faces, each one an equilateral triangle.

A cube, or regular hexahedron, has six square faces (below).

A regular octahedron has a surface consisting of eight equilateral triangles (below).

A regular dodecahedron is made up of twelve regular pentagons. If you were to cut it into two equal parts, each part would resemble a flower having five pentagon-shaped petals around a central pentagon (below).

A regular icosahedron has twenty flat surfaces, each one an equilateral triangle (below).

More than two thousand years ago, the Greek mathematician Euclid proved that these five objects are the only ones that can be constructed from a single type of regular polygon. Known as the Platonic solids, they are named after the Greek philosopher Plato, who lived around 350 B.C.

The Greeks were not the first to study these shapes, however. There is evidence that people in China and in the British Isles knew about them long before Plato's time.

Plato believed that the world was made up of tiny particles consisting of four elements: fire, air, water, and earth. Each particle had the shape of a regular polyhedron.

Fire, the lightest and sharpest of the elements, was a tetrahedron. Earth, as the most stable element, consisted of cubes. Water, as the most mobile, was an icosahedron, the regular solid most likely to roll easily. Air was an octahedron, and the doecahedron represented the entire universe.

Discover an amazing relationship among the vertices, edges, and faces of polyhedra.

You will need:
  • sheet of paper, pencil, ruler
  • cube-shaped object, such as a game die, a child's building block, or a box
  • any other examples of polyhedra that you can find, such as dodecahedral, octahedral, or tetrahedral dice (used in certain games) or objects like prisms
  • soccer ball
What to do:
  1. Divide your sheet of paper into four columns.
  2. Label the head of each column, in order,  Name of Shape; Faces (F); Vertices (V); Edges (E).
  3. Under "Name of Shape," write CUBE.
  4. Count the number of faces on the cube and record your total under "F."
  5. Count the number of vertices on the cube and record your total under "V."
  6. Count the number of edges on the cube and record your total under"E."
  7. Record the same information for each polyhedron you have available. For a soccer ball with a surface pattern of hexagons and pentagons, think of each face as a flat rather than a curved surface.
  8. Look for patterns in the table to find a relationship among the number of faces, vertices, and edges. Hint: Try adding or subtracting various combinations of F, V, and E for each polyhedron.
The relationship you are looking for was discovered by Leonhard Euler, an eighteenth-century Swiss mathematician. The rule applies to many different kinds of polyhedra. Use it, for example, to calculate how many edges a solid with eight faces and twelve vertices must have.

The relationship is F + V − E = 2.
If F = 8 and V = 12, then F + V = 20, so E = 18. That means a polyhedron with 8 faces and 12 vertices has 18 edges. It is called a hexagonal prism.

NEXT: The Amazing Buckyball

July 17, 2020

The Buckyball Asteroid

3. The Buckyball Asteroid

You are down on your knees fitting a bunch of triangles and squares into an awesome pattern when you look up and spot a girl wearing a baseball cap and a white soccer shirt with the number "21."

"Hi. What's happening?" she asks casually, as if the two of you were back home, not on some asteroid.

"Just tiling the floor," you answer.

"I'm Anita," she says. "My friend Bill over there is looking for Buckyball Field," she adds, pointing to a long-legged, curly haired boy who is kicking a soccer ball. His white soccer shirt has the number "34."

"Did you find it?" Bill asks Anita as he comes running up. "I know this sounds crazy," he explains, "but we're scheduled to play soccer against some space aliens at Buckyball Field, and we don't know where it is."

"I think we're on the wrong asteroid," Anita says with a sigh.

"I bet we could find it from my space capsule," you offer.

You, Anita, and Bill are soon hanging out in what used to be your bedroom, silently sailing through space.

A sharply pointed shape suddenly appears on the navigation screen. Looking closely, you notice that it has four identical triangular faces.

Then other three-dimensional objects come into view. Some have triangular surfaces, like the first one you saw, and others are made up of squares, pentagons, or other polygons.

"Look! There's one made up of twenty triangles," Anita says.

"I see a buckyball!" Bill exclaims, pointing to an object on the screen that looks a bit like a soccer ball. "Quick! Click on it!"