Tiling
Min Yan, Department of Mathematics
Uruk, Sumer (Iraq), 3400-3100BC
Tomb of King Djoser, Egypt, 2668-2649BC
Chogha Zanbil, Elamite (Iran), 1250BC
Glazed brick tile
Chinese Bronze, ~1000BC
Ishtar Gate, Babylon (Iraq), 575BC
Pergaman Museum, Berlin
Gr

The Man Who Knew Innity and other Functions
Edmund Y. M. Chiang
Captsone Project presentation
19th November 2015
A lm
Early Life
G. H. Hardy
Rogers-Ramanujan identities
Some qcalculus
Outline
A lm
Early Life
G. H. Hardy
Rogers-Ramanujan identities
Some qc

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Perspective Drawing
and Projective Geometry
Capstone
WeiPing Li
The Renaissance was a cultural movement that
profoundly affected European intellectual life in
the early modern period (15th century).
Beginning in Italy, and spreading to the rest of
Europe

ELLIPTIC FUNCTIONS AND THETA FUNCTIONS
LECTURE NOTES FOR NOV.24, 26
Historically, elliptic functions were rst discovered by Niels Henrik
Abel as inverse functions of elliptic integrals, and their theory was improved by Carl Gustav Jacobi; these in turn we

POINCAR-BENDIXSONS THEOREM: APPLICATIONS AND THE PROOF
FREDERICK TSZ-HO FONG
This lecture note is written for the two lectures of MATH 4999 (Fall 15) in Week 4. The
content is extracted from a chapter of lecture notes written by the author for a course on

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The Problem of the Thirteen Spheres
One can easily arrange 12 unit spheres all touching a central
one:
For example, touching the central one at the 12 vertices of
an inscribing regular icosahedron.
1
Note: This arrangement is very untight.
tan1 2
4 sin
=

Spherical Tiling by Congruent Pentagons
Min Yan
November 12, 2015
webpage for further reading:
http:/www.math.ust.hk/mamyan/research/UROP.shtml
We consider tilings of the sphere by congruent pentagons. The basic
example is dodecahedron, which has 12 penta

Contractive Mapping Theorem
Kin Li
We will introduce an important theorem that has applications in many areas of mathematics, including analysis, linear algebra, dierential equations, dierential geometry and
applied mathematics. To begin, let us consider

Archimedes formula: A = 4R2
R
2R
From now on, take R = 1.
1
Localization I: area of a spherical lune = 2
Localization II (Girards Theorem): The area of a spherical triangle ABC on the unit sphere is:
= A + B + C .
2
A
B
C
C
B
A
ABC + A BC = 2 A
ABC + ABC

Hausdor Metric
Kin Li
Here we will comment more on the last example of the contractive mapping theorem
notes. In the proof of the contractive mapping theorem, we need to use the triangle
inequality and the property d(x, y) = 0 implies x = y to have unique