The DC Math Trek begins at the Judiciary Square Metro station in downtown Washington, D.C.
Note the hexagonal tiles on station platforms and in other areas. In some stations, these tiles are being replaced by rectangular blocks with an inscribed hexagonal pattern. However, the patterns on adjacent tiled rectangles do not form a precisely hexagonal tiling when block edges are matched to form hexagons.
See "From Hexagon to Square."
Exit the station to face the National Building Museum (401 F Street, N.W.).
The building's facade illustrates the rampant use of symmetry in its architectural design.
Frieze patterns with repeated designs decorate many of the building's exterior walls.
Enter the building via the main entrance to see more varieties of symmetry in the grand, colonnaded atrium and the carpeting on the main floor.
Columns in the grand atrium of the National Building Museum.
The carpeting in the grand atrium of the National Building Museum features a symmetric pattern.
Leave the building and walk east along F Street, N.W., to the corner of 4th Street, N.W., and turn right, proceeding south on 4th Street, N.W.
Observe the recycling receptacles at the corner of F and 4th (and elsewhere along the route) and the recycling symbol of three bent arrows that they display.
The recycling symbol here has two identical arrows and a third bent in the opposite direction. Elsewhere you may see a mutant variant that consists of three identical arrows--also representing a one-sided, one-edged Moebius strip but with three twists instead of just one.
See "Recycling Arrows."
Take a close look at the next fire hydrant you see (such as the one at 4th and D).
Notice that the nut atop the hydrant is pentagonal. Why would it be pentagonal rather than hexagonal like the nuts holding the hydrant's upper part in place?
See "Fire Hydrant Pentagons."
Cross D Street and move slightly to the left (east) to reach a park and walkway between buildings. The park features an eye-teasing example of geometric, minimalist art.
See also "Solid Illusion."
The walkway taking you to C Street has several distinctive pavement patterns, each having a different symmetry.
One tiling features squares and elongated hexagons to create an octagonal pattern (above). Another shows a weaving of rectangles (below).
Go to the right (west) to reach another pedestrian walkway (John Marshall Memorial Park), with the Embassy of Canada on the right.
The Embassy of Canada, designed by architect Arthur Erickson, features long horizontals and wide open spaces to evoke a sense of Canada.
Note the sharp corners (both acute and obtuse) of the building, which is based on a trapezoid split into two triangles.
See "Splitting a Trapezoid."
The building's Central Court contains a large Alexander Calder hanging mobile, visible against the ceiling's triangular motif.
An untitled mobile sculpture by Alexander Calder hangs from the ceiling of the National Gallery of Art's East Building.
Exit the East Building and go to its remarkable southwest corner, where two walls meet at 19 degrees.
The walls creating the sharp corner farthest to the right meet at 19 degrees.
Look closely at the discoloration of the marble at about shoulder height on the sharp corner..
The smudge shows where thousands of people have over the years touched the corner, staining the marble and producing a pattern that can be construed as a population distribution.
See "An Irresistible Edge."
Cross the plaza to the east entrance of the National Gallery of Art. Proceed into the building, then down a few steps. On your right in the hallway (which leads to the Gallery Shop), you see The Sacrament of the Last Supper by Salvador Dali, with its dodecahedral background.
The National Gallery of Art houses Salvador Dali's The Sacrament of the Last Supper, with its mystical dodecahedral background.
Exit the building by the same entrance as you entered and continue along 4th Street across the National Mall, with views of the U.S. Capitol to the left and the Washington Monument to the right.
Looking west from 4th Street along the National Mall toward the Washington Monument.
Cross Jefferson Drive, S.W. On the left, you see the curvy facade of the National Museum of the American Indian.
The golden-hued building housing the National Museum of the American Indian is clad with Kasota limestone. Its curvilinear form evokes natural rock formations shaped by wind and water.over thousands of years.
To the right you see the boxy forms of the National Air and Space Museum.
The National Air and Space Museum's building consists of four cubes encased in marble and connected by three glass-and-steel atria.
Walk south along 4th Street to Independence Avenue, then turn right to go west along Independence Avenue to the National Air and Space Museum's main entrance. Near the entrance you'll see Charles O. Perry's sculpture Continuum.
Continuum by Charles O. Perry is modeled on a Moebius strip, albeit a very twisty one.
See "Moebius Contiuum."
Continue west along Independence and cross 7th Street to the main entrance of the Hirshhorn Museum and Sculpture Garden. Keep to the right of the building to see the outdoor sculptures, especially Kenneth Snelson's Needle Tower.
Needle Tower by Kenneth Snelson is a wonderful example of a tensegrity structure. In this cable-and-rod form, none of the rods is connected directly to any of the others.
See "Needle Tower."
If you step underneath Snelson's tower and look upward, you'll see the sculpture's striking sixfold symmetry.
Continue through the Hirshhorn Museum's courtyard to the museum's Jefferson Drive exit. Turn right and walk east along Jefferson to 7th Street. Turn left and go north on 7th Street back across the National Mall to the National Gallery of Art Sculpture Garden. The Sculpture Garden's east entrance is at 7th Street and Madison Drive, N.W.
Southeast entrance to the National Gallery of Art Sculpture Garden.
Four-Sided Pyramid by Sol LeWitt is made up of rectangular blocks.
See "LeWitt's Pyramid."
To the west of LeWitt's pyramid, you'll see Moondog by Tony Smith.
Moondog by Tony Smith has a structure based on a regular octahedron and is composed of .15 octahedra and 10 tetrahedra.
During the summer, the water sprays spilling into the large pool to the north near the center of the sculpture garden provide a nice example of parabolic arcs.
Water sprays create graceful parabolic arcs.
See "Fountain Parabolas."
Leave the sculpture garden via its west exit and walk along Madison Drive past the National Museum of Natural History, with a view of the Smithsonian "castle" across the mall to the left, until you reach the main entrance of the National Museum of American History. Note the large rotating sculpture mounted on a pedestal in front of the building.
Infinity by Jose de Rivera is a three-dimensional analog of a Moebius strip, with triangular cross section.
See "Infinity in Eight Minutes."
Continue along Madison Drive, cross 14th Street, and head toward the Washington Monument.
With its square cross section and pyramidal top, the Washington Monument is the world's tallest obelisk.
On the grassy grounds surrounding the obelisk, about 100 yards to the northwest, you'll find a stone marker commemorating the Jefferson Pier.
The Jefferson Pier marks the second prime meridian of the United States.
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). The Jefferson Pier stands at the triangle's right angle.
The Jefferson Pier stone marker, with the White House due north and the Thomas Jefferson Memorial due south, lies 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. 16th Street N.W., follows the path of the meridian.
Proceed northward across the mall toward the White House to reach Constitution Avenue. The Ellipse is a grassy area, encircled by a driveway, in front of the White House.
Is the field (and driveway) known as The Ellipse truly elliptical? What measurements could you make on the ground to confirm whether the site does (or does not) meet the mathematical definition of an ellipse.
Turn left and walk eastward along Constitution Avenue to 14th Street, N.W. At the northwest corner of 14th and Constitution, you'll find a Bulfinch Gatehouse.
A braid pattern decorates the Bulfinch Gatehouse and other structures near and on the National Mall.
See "Loops, Knots, and Unknots."
Proceed north along 14th Street to E Street, N.W. Turn right and walk west along E Street, taking you to the north side of The Ellipse, across from the White House, where you find another meridian marker (the Zero Milestone) and the National Christmas Tree.
The view from the Zero Milestone and prime meridian marker northward toward the White House.
The National Christmas Tree's branched (somewhat fractal) structure contrasts with the Euclidean straight-line purity of the Washington Monument in the background.
See "Skylight Fractal."
Return to 14th Street and walk northward. You pass the venerable Hotel Washington on the right, with its ornately decorated facade.
Hotel Washington (now W Washington D.C.).
The neoclassical U.S. Department of the Treasury Building is on the left as you walk northward on 14th Street, N.W..
At Pennsylvania Avenue, turn left to get a view of the White House and of the Andrew Jackson statue in Lafayette Square, on the right.
An equestrian statue of Andrew Jackson stands in Lafayette Square in front of the White House. The Washington Monument is visible in the background.
At the north end of Lafayette Square you can look northward along 16th Street, N.W.
16th Street, N.W., stretches northward along what was once the prime meridian of the United States.
Walk along 16th Street to K Street, N.W., turn left and follow K Street westward to reach the Farragut North Metro station.
What other sites/sights of mathematical interest did you see on this D.C. math trek?
Photos by I. Peterson