Lighthouse – Mathematical Connections

The story

The lighthouse  illuminates the world around it, enabling seafarers to safely navigate unknown waters.

Look closely; what do you see?

Atop the lighthouse are two light sources that cast light and shadows. Can you find…

• two spots of light on the walls–one orange, one yellow?
• a geometric pattern of white light and dark shadow on the ceiling?

Focus on… stereographic projection

Stereographic projection uses light to cast shadows through a sphere onto a plane. The planar surface here is the ceiling.

Stereographic projection preserves some geometric properties while distorting others. For example, the angles of triangles on the sphere are preserved in the shadow (so ninety degree angles on the sphere project onto ninety degree angles in the shadow). Lengths are not preserved: lengths in the shadow increase the farther the plane is from the sphere.

The mathematics of projection has social and political implications. The Mercator projection is a way to map continents from a globe to a flat plane–but land further from the equator becomes disproportionately large. How does this affect how we perceive our world?

In this Mercator projection map of the Earth, Greenland appears almost as large as Africa, and Antarctica dwarfs both. In reality, Africa has over 14 times the area of Greenland and twice the area of Antarctica.

Lighthouses have become mostly obsolete due to satellite navigation. Note that both types of navigation (by triangulating the lights of nearby lighthouses, or by using Global Positioning Satellite or GPS signals) rely on quite a bit of mathematics!

The lenses used in the Mathemalchemy lighthouse are Fresnel lenses, just as in a real lighthouse. Fresnel lenses were developed in the 19^th century by French physicist Augustin-Jean Fresnel; they make it possible to make powerful lenses using much less refracting material, thereby allowing physically realizable light beams with much greater focus, visible from greater distances.  The Fresnel lens has been called “the invention that saved a million ships”.

One of the organizing principles of the Lighthouse is the regular heptagon. The heptagon plays a special role in the history of mathematics, because it is the smallest regular polygon (i.e. the one with fewest sides) that can not be constructed with a straightedge and a compass; before this was proved, many people tried to find a way …

The lighthouse was designed mathematically to make its welding especially easy; this required a precise matching of angles in the design of the different curves. The climbing spiral of the outside ramp and the slight “vase” shape (with a more narrow “waist” in the middle than at the top and bottom) needed to be paired with a less obvious spiraling of the (almost) vertical metal strips to make this happen.

More precisely, the local rotation of each curve was decomposed into the rotation around the normal (geodesic curvature) and around the binormal (normal curvature). Now imagine welding together two strips of metal, with a “T” shaped cross-section. The strip corresponding to the top bar of the “T” is cut from a flat sheet according to the geodesic curvature of the original curve; the strip corresponding to the vertical line of the “T” is likewise cut from a flat sheet, contributing the normal curvature. Bending and welding the two strips together essentially integrates these two curvatures to give back the full original curve. This was the principle according to which the components of the lighthouse were designed, with only the slight difference that the spiral ramp used an “L” rather than “T” shape.

The light beam projecting unit is housed in a dodecahedron.  The dodecahedron can be viewed as referring back to the 5-fold symmetry present in many ways in the nearby Bakery. But it also has very special mathematical properties that single it out from among the Platonic polyhedra, making it appropriate as a polyhedron choice for the building that rises above everything else in the installation.

One of these properties, illustrated here, was discovered only recently. In a straight-line path on a polyhedron you walk straight while on a face; at every edge crossing, you continue on the new face so that your path makes the same angle on your right with the common edge as it did on your left on the previous face. (You could also imagine the common edge as a hinge; if you were to imagine the new face hinged around until it lay in the same plane as the old face, but on the other side of the hinge, then your path would be a straight line with no deviation.) If you walk in a straight-line path on a Platonic solid other than the dodecahedron, starting from a vertex and aiming to return to that same vertex, then you will always travel through another vertex before you are back at your starting point. Not so on a dodecahedron – Jaydev Athreya, David Aulicino and Patrick Hooper showed examples and classified all possible such other-vertex-avoiding straight paths. The boundary between yellow and red on the Lighthouse dodecahedron follows one of the straight paths they constructed.

Stereographic projection is an example of a fascinating group of transformations, called conformal mappings – these are geometric transformations that preserve angles (but not lengths). A fun example using stereographic projections in a mathematical video is here.

Another well-known conformal map is the Mercator projection, used for many geographic maps.