Mathematics abounds in the world around us. Here are more
than a dozen sights (sites) at or near the National Mall in Washington, D.C., where
you can catch glimpses of triangles and trapezoids, knots and Möbius strips,
fractals and pyramids, and more. Join us for a mathematical treasure hunt among
the monuments, museums, and fields at the heart of the nation's capital.

**Splitting a Trapezoid**

The National Gallery of Art's East Building is characterized
by vast walls that meet at oblique angles. It sits on a four-sided site shaped like
a trapezoid, with one side running diagonally. Architect I.M. Pei came up with
his design for the building by splitting the trapezoid into an isosceles
triangle and a smaller right triangle, creating an abundance of acute and
obtuse angles. Triangles serve as the structure's basic motif.

For more, see the blog post "Splitting a Trapezoid."

**An Irresistible Edge**

The National Gallery of Art's East Building has one
particularly sharp corner, visible to the right of the the building's H–shaped,
west-facing façade. There, two walls meet at 19° to form the apex of a narrow
triangle. If you look closely at the corner, you'll see a smudge darkening the
lavender-pink marble. Over the years, so many people have touched the corner that
their hands have collectively left a distinctive mark. It stretches about two
feet, tapering off at its upper and lower ends. In effect, it is a population distribution,
representing all the people who have reached out to the enticing corner.

For more: "An Irresistible Edge"

**Hexagons and Pentagons**

Regular hexagons fit together without gaps to cover a
surface, as seen in the floor tiles of any MetroRail station. Such hexagons
have three pairs of parallel sides, a handy feature for nuts and bolts that can
be tightened or loosened using a standard wrench. Regular pentagons, on the other
hand, don't fit together neatly to cover a surface completely, and such pentagons
have no parallel sides. A fire hydrant has pentagonal lug nuts, which make them
wrench-proof.

For more: "Fire Hydrant Pentagons"

**Fractal Branches**

The straight lines of the Washington Monument contrast
sharply with the intricately branched structure of the National Christmas Tree.
Nature is full of shapes that repeat themselves on different scales within the
same object—branching plants, puffy clouds, rugged coastlines, craggy
mountains, and jagged lightning bolts. Mathematician Benoit Mandelbrot coined
the term "fractal" as a convenient label for such self-similar shapes, which
are difficult to describe using conventional geometry.

For more: "Skylight Fractal"

**LeWitt's Pyramid**

Sol LeWitt's

*Four-Sided Pyramid*at the National Gallery of Art Sculpture Garden consists of concrete blocks stacked to form a stark, eye-catching pyramid. Although the blocks are rectangular—each one equivalent to two cubes attached side by side—LeWitt's structure can look like a huge pile of cubes from some viewpoints. At the same time, the pyramid is highly symmetrical. Can you use that symmetry to help calculate how many blocks make up the pyramid's faces without counting all of them?
For more: "LeWitt's Pyramid"

**Needle Tower**

Erected beside the Hirshhorn Museum and Sculpture Garden,
Kenneth Snelson's

*Needle Tower*stretches 60 feet into the sky. This tapered framework of aluminum tubes and stainless-steel cables is an example of a tensegrity structure. The tubes aren't connected to each other. Instead, cables connect the tubes to hold the assemblage together in perfect balance. When you venture underneath and look up, you see a striking pattern of six-pointed stars, a consequence of the fact that each layer of a tensegrity structure must consist of three tubes.
For more: "Needle Tower"

**Möbius Continuum**

A complicated, twisty form stands guard at the entrance to
the National Air and Space Museum. Created by architect and sculptor Charles O.
Perry, the sculpture is called

*Continuum*. Perry describes his artwork as a Möbius surface of seven saddles, designed to represent the continuum of the universe. In effect, despite its confounding twistiness, the sculpture has just one continuous surface and one edge—just like a Möbius strip.
For more: "Möbius Continuum"

**The Washington Right Triangle**

Washington, D.C., was planned around a right triangle, with
the White House at its northern vertex and the U.S. Capitol at its eastern
vertex, joined by Pennsylvania Avenue (as the hypotenuse). A stone marker (shown above), commemorating
the Jefferson Pier, stands at the 90° vertex, a short distance north and west
of the Washington Monument. The marker, with the White House due north and the
Thomas Jefferson Memorial due south, is on what was once the prime meridian of
the United States. An equestrian statue of President Andrew Jackson stands on the meridian
north of the White House.

For more: "The Washington Right Triangle"

**Recycling Topology**

It's hard to miss the triangle of three bent arrows that
signifies recycling. You can find it on bins and packaging, in magazines and
newspapers, and elsewhere. Created in 1970, the design was based on a Möbius
strip to symbolize continuity within a finite entity. Have you noticed,
however, the two versions of this symbol? In its original form, just two of the three
arrows have the same twist, while in the mutant form all three arrows twist the
same way. Does this mutant form also have the one-edged, one-sided surface characteristic
of a Möbius strip?

For more: "Recycling Arrows"

**Water Fountain Geometry**

Whether spraying graceful arcs of water into the air or
letting water tumble down steep slopes, outdoor fountains draw attention. One factor that makes some
fountains more spectacular than others is the angle of the jets that send water in parabolic
paths. Angles between 50° and 60° seem to produce the most spectacular sprays, either
enclosing the largest possible volume or having the greatest total surface area. The fountain at the
National Gallery of Art Sculpture Garden is a notable example.

For more: "The Geometric Spectacle of Water Fountains"

**Infinity in Eight Minutes**

Eight feet tall and 16 feet wide, a stainless-steel loop
revolves once every eight minutes atop a black granite pillar in front of the National Museum of American History.
Designed by José de Rivera, who titled the piece

*Infinity*, the looped sculpture is a three-dimensional analog of a Möbius strip. The loop's cross section is an equilateral triangle, and this triangle rotates through 120° before the ends meet to form a complete loop. Instead of three surfaces, the final product has just one continuous surface that runs three times around the loop.
For more: "Infinity in Eight Minutes"

**Knot or Unknot**

The carved stone decorations on the Bullfinch Gate House, located
near the White House, look like mathematical knots. Mathematically, a knot is a
one-dimensional curve that winds through itself in three-dimensional space and
catches its own tail to form a loop. If the loop has no knot in it and can be
made tangle-free to look like a circle, mathematicians call the loop an unknot.
Are the Gate House designs true knots or merely disguised unknots?

For more: "Loops, Knots, and Unknots"

**Manhole Cover**

Manhole covers are usually circular and often feature geometric designs. It's rare to find one that is 12-sided (dodecagonal), like some of the manhole covers near the U.S. Capitol. These covers also feature a hexagonal grid and a curious pattern of holes.

For more: "Manhole Cover Geometry"

**Unit Signs**

Signs at street corners in Washington, D.C., give not only
the street name but also a numerical indicator representing the numbers
assigned to a given block—typically in multiples of 100. As you approach the
U.S. Capitol, you see signs, block by block, saying 600, 500, 400, 300, 200,
and 100. Then what happens? Would the signs say 000? Instead, city planners
chose the designation “unit” to mark blocks with street numbers from 1 to 99.

For more: "Unit Signs"

See the MAA's page on "A Field
Guide to Math on the National Mall" to download a brochure
(PDF) featuring these fourteen sights.

Photos by I. Peterson