More than Meets the Eye Geometry and Our

More than Meets the Eye: Geometry and Our Perception of Reality Richard G. Ligo The University of Iowa

Overview • Introduction • The shape of the Earth • Determining Earth’s size • Making maps of the Earth • Curvature applied to reality • The shape of the Universe

Hints of the Earth’s shape Lunar eclipses:

Hints of the Earth’s shape Horizon of the ocean:

Hints of the Earth’s shape Constellation visibility:

Hints of the Earth’s shape Eratosthenes and the gnomon

Eratosthenes and the gnomon

Mapping the Earth Theorema Egregium (Gauss) The Gaussian curvature of a surface is invariant under isometries. Intuitively, theorem says that a surface may be “bent” without stretching or squishing it and have the same Gaussian curvature.

Mapping the Earth Definition: A surface is called developable if it has zero Gaussian curvature.

Maps: central stereographic projection

Maps: azimuthal equidistant projection

Maps: central cylindrical projection

Maps: equirectangular projection

Maps: Lambert cylindrical projection

Maps: Mercator projection

Derivation of the Mercator projection ?

Derivation of the Mercator projection Globe Projection

Derivation of the Mercator projection

Derivation of the Mercator projection

Maps: Natural Earth projection

The curvature of a surface K=0 K<0 K>0 K? 0

The curvature of a surface K=0 C = 2πr

The curvature of a surface K>0 C < 2πr

The curvature of a surface K<0 C > 2πr

The curvature of space

The shape of the universe

The shape of the universe

The shape of the universe

The shape of the universe

The shape of the universe

The shape of the universe ?

References • Stewart, Ian (2001). Flatterland. Cambridge, MA: Perseus Publishing. • Oprea, John (2007). Differential Geometry and Its Applications. Washington, DC: Mathematical Association of America. • Osserman, Robert (1995). Poetry of the Universe. New York, NY: Anchor Books.