It was already known to some people four thousand years ago that the triangle with edge lengths 3, 4 and 5 is a right-angled triangle. The Pythagorian Theorem asserts that for a right-angled triangle, the sum of squares of two legs is equal to the square of the hypotenuse. This theorem gives a relationship between figures and numbers; however, some more time was necessary for people to understand the fact in this way. This understanding should have opened the ways to design things that the society wants.

Streets in your town, each building there, each furniture in your room, the personal computer on your desk, books, your cellphone, … that you see every day are all designed using nice properties of the figures. Because planar polygons such as triangles and quadrilaterals form the faces of three-dimensional figures, it was necessary to understand their properties well.

In Euclid’s *Elements* written over two thousand years ago, deductive theories of figures and quantities based on axioms are developed. Our modern life is founded on the basis of these rigorous theories on figures. The world of three-dimensional figures is so rich that new phenomena continue to be discovered today.

This lecture series addresses how properties of the figures have been understood and how that understanding is applied to the real world, and explains the recent social requests and the development of modern mathematical science on figures to respond these requests. A grasp of high school mathematics is sufficient background knowledge to take this course.

### Polyhedra

#### Lecture 1 4/7 Euler’s Formula and The Regular Polyhedra

#### Lecture 2 4/14 Rigidity and Deformation of Polyhedra

#### Lecture 3 4/21 Unsolved Problems on Polyhedra

##### Masahiko Kanai (Faculty of Science)

Probably you feel that you know very well about POLYHEDRA, surrounded by a plenty of them in your daily life. Surprisingly enough, though, there are quite a few open problems on them, which are stated in purely elementary words so that even small children can understand what those problems mean. These three lectures are designed as a rather informal or casual introduction to the subjects. Indeed, we are planning to do even some handicrafts. We wish you to enjoy the class playing with polyhedra.

### Mathematical Principles and Computations of Origami

#### Lecture 4 4/28 Computational Origami and Shapes

#### Lecture 5 5/12 Mechanisms of Origami

##### Tomohiro Tachi (College of Arts and Sciences)

Origami, the traditional Japanese art of folding paper, has now become a hot research theme studied worldwide across various fields such as art, mathematics, computer science, material science, life science, structural engineering, robotics, and architecture. These lectures will introduce mathematical principles and computational techniques of origami that bridge these interdisciplinary themes, illustrating design systems for creating almost all kinds of shapes in origami, mechanisms of folding paper, and designs for hard yet soft origami structures.

### Periodicity and Symmetry

#### Lecture 6 5/19 Recognizing Symmetry, Recognizing Periodicity

#### Lecture 7 5/26 Understanding Space with Rotations and Translations

#### Lecture 8 6/9 Polyhedra That Fill the Space

##### Takashi Tsuboi (Faculty of Science)

We can discover many symmetric figures in everyday life and in nature. Numerous artworks possess beautiful symmetry as well. Periodicity can be observed in a multitude of infinitely expanding figures. We can describe symmetry and periodicity in the space by mathematics. Then we can classify the symmetry or the periodicity into finitely many groups.

### Polygon and Polyhedron with Simulations

#### Lecture 9 6/16

##### Norikazu Saito (Faculty of Science)

Various phenomena such as a tsunami run-up are simulated using computers. Objects of computations such as structures have complex shapes in general, but are regarded as small collections of triangles and tetrahedra to represent shapes. This lecture will examine good shapes of triangles and tetrahedra from the perspective of simulations.

### Interactive Shape Modeling

#### Lecture 10 6/23

##### Takeo Igarashi (Faculty of Science)

This lecture will introduce techniques of shape modeling that form the basis of computer graphics used for games and movies, and CAD (computer-aided design) used for industrial products and architecture. In particular, we will discuss techniques for creating sophisticated graphical representations such as three-dimensional shapes and shapes that meet specific physical properties by using as simple manipulations as possible.

### Looking at the Shape of Space from Four-Dimensional Polytopes

#### Lecture 11 6/30 Let’s Look at Four-Dimensional Figures

#### Lecture 12 7/7 What Does It Mean That Space Is Curved?

#### Lecture 13 7/14 Topology of the Universe

##### Toshitake Kohno (Faculty of Science)

We can project three-dimensional figures on a plane, but how can we visualize four-dimensional figures? In these lectures, we will first consider regular polyhedra in the four-dimensional space. Second, the lecture will explain the history of space recognition from Euclid’s *Elements* to the discovery of non-Euclidean geometry and Riemannian geometry. Furthermore, the lecture will describe approaches to “the shape of the universe” based on the findings of modern cosmology and geometry.