Can you tell the shape
from numbers ?
Platonic Solids
Platonic solid is a convex, regular
polyhedron in 3D. Being a regular
polyhedron means that the faces are
congruent (identical in shape and size)
regular polygons (all angles congruent
and all edges congruent), and the
same number of faces meet at each
vertex. There are only five such
polyhedra
The Euler Characteristic Theorem
For any connected graph (also called network) in the plane,
V –E+F=2
where V is the number of vertices, E is the number of
edges, which divide the plane into F regions (counting the
region outside the graph as an additional one !)
Polyhedrons
The Euler Characteristic Theorem
If a Polyhedron can continuously deform into a sphere,
then V –E+F=2, where V is the number of vertices, E is the
number of edges, and F is the number of faces
A bear hunter sets out from
camp and walks one mile south.
He sees a bear and is about to
shoot it. The bear grabs his gun
and eats it. The hunter runs
away one mile east. He then
walks one mile north and gets
back to his camp.
Trip on the earth
The shortest path connecting two points on a
sphere is the great circle: the intersection of the
sphere and a plane that passes through the center
point of the sphere
Triangle on the sphere
The sum of angles for a triangle on the
sphere is greater than 180 degree !
Mercator projection
USA：9.834 million km²
Greenland：2.166 million km²
Measure the world
Gauss’1827: the Earth, or part of it, cannot be
displayed on a map without distortion.
Gauss Curvature
Positive curvature if sum of angles
exceeds 180 degree, like an egg.
Zero curvature if sum of angles equals
180 degree.
negative curvature if sum of angles
under 180 degree, like a saddle.
Gauss: On any surface, the distance between
any pair of locations will determine a metric.
Then an accurate map would require a metric
that applies to the map and the sphere !
Gauss theorem: The metric on the surface
determines its Gauss curvature !
Arched structure is more stable
Theorema Egregium: it is
impossible to bend a
plane though two
perpendicular directions
without fractures.
Curvature of a plane curve
Curves on a surface
At each point p of the surface, one can
find directions with max/min curvature,
called principle curvatures k1 and k2. The
Gauss’s curvature is defined by K=k1 * k2 .
Gauss-Bonnet theorem: the integral of the
Gaussian curvature over a surface
depends only on the number of holes in
that surface
Thank you for your attention