1. January 9 Lecture (preview)

2. January 14 Lecture (Ruler-compass constructions)

3. January 16 Lecture (The Golden Section)

4. January 23 Lecture (Fibonacci numbers)

5. January 28 Lecture (Symmetries)

6. February 4 Lecture (Tilings)

February 6 Lecture (Fractals).

1. Here is an interactive tree fractal that we have already seen (January 16 lecture) - as you might recall, the "best" tree is obtained if we choose the ratio of the consecutive branches to be 1/Golden Ratio.

2.An interactive page where you can explore 5 fractals: Koch snowflake, Dragon curve, a tree fractal depicted in the textbook, a "crystal" also seen in the textbook, and the Hilbert curve.

3. A gallery of beautiful fractals - some are home made, some are scooped from the web: these are related to the Julia sets (associated with some special rearrangements of the points in the plane which are neither similarities nor isometries). Details explained in class.

(a) Here is the "basic Julia (Mandelbrot) fractal.

(b) We start with a another picture of the Julia fractal. First we zoom in once. We see a "caterpillar". Here is what happens if we zoom in a bit around one of the legs of the "caterpillar". We zoom in more. And more. And once again for the last time. We see that the "legs" continue to pop up, smaller and smaller.

(c) Some beautifull fractals; it is math that generates them.

Under the sea	Music	Week
Eden	Flowers 1	Flowers 2
Face	Sea Shell	Space
Tree	Clown	Flower 3

For the Sierpinski gasket and Menger sponge go to the Animation Gallery.

February 13 Lecture (Review for Midterm).

1.Here is the midterm exam from the fall of 2002 in pdf. Otherwise, you can see page 1 and page 2 separately.

2. A list of suggested questions form the texbook:

Pages 90-93: 1, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14.

Page 95: 1, 2, 4, 5, 6, 7, 8

Page 96: All questions on that page

Pages 104-105: all questions on these two pages (fractals).

3. Here is the midterm from the Fall of 2001 (in pdf - you need the Adobe Acrobat Reader to see it). Please note that both 2001 and 2002 midterms were based on the material covered during the corresponding sessions - that material is not entiraly equivalent to what we have done so far, and, moreover, we have covered some parts differently. So, please do take a look at the questions given in the old midterm exams, but do not overestimate their similarity to what you will see in your own midterm exam.

February 15 Lecture (Perspective).

1.Objects far away appear smaller.

Picture 1.

Picture2.

Picture3.

Picture 4.

2. A picture from the pre-renaissance (in the discussion board - link up).

3. Renaissance - early attempts: Jean Fouquet, 15th century.

4. One of the early masters who knew perspective: Albrecht Durer (15th/16th century).

4. Leonardo da Vinci.

5. A small analysis of Jean Fouquet.

6. Playing games with perspective.

(a) Ascending and Descending by Escher (1960).

(b) Ascending and Descending again, with a bunch of lines. The trick is hiden in a spiral.

(c) Relativity by Escher (1953).

(d) Carefully-restored Roman ruin in a forgotten Flemish locality with Oriental influences by Jos de Mey (1983).

March 6 Lecture (String Art).

1. Here is the standard string-art picture of a parabola.

Two such parabola and ... an "eye". Here are a bunch of parabolas in a cuter picture.

2. Playing with a cardoid: a sript for constructing hearts with strings.

March 13 Lecture (Regular Polyhedra).

1. Artists from the renaissance :

(a) Polyhedra, Luca Pacioli 1445-1514

(b) Intarsia Polyhedra , Fra Giovanni c.1520

(c) Geometria et Perspectiva 4, Lorenz Stoer, 1567

2. The ubiquitous Escher:

(a) Do you recall the reptiles ?

(b) A tetrahedron (Tetrahedral Planetoid, 1954)

(c) A double tetrahedron (Double Planet, 1949)

(d) Find Waldo (Stars, 1947)

3. A script with all of the regular solids (as interactive 3D graphics) and many other polyhedra is accessible through the webMath page for this course.

March 18 Lecture (Hyperbolic Plane).

1. The sphere in the above picture does NOT represent a model of the hyperbolic plane. It is here to illustrate two aspects of the Poincare model of the hyperbolic plane: the lines are not "straight" and the distances are not the "usual" or Euclidean distances. Details to be explained in class.

2. Escher again: the picture is from his early period, and it is called "Hand with reflecting globe" (1935)

March 25 Lecture (Hyperbolic Plane and Tilings).

1. Escher:

(a) Circle Limit 1, Escher, 1958

(b) Circle Limit 3,Escher, 1959, and the associated (4, 3, 4, 3) hyperbolic tiling. Here we see both of them side by side.

(c) Circle Limit 4 Also known as Angels and Devils (Escher, 1960)

2. A script for drawing nice hyperbolic tilings (accessible through the webMath page for this course.)

3.There is a 3D analogue of the hyperbolic plane: the hyperbolic space. Here is a sculpture inspired by the hyperbolic space.

March 27 Lecture (Topological structures).

1. Can you deform the 3 dimensional object behind this link into this one WITHOUT cutting or pasting?

2. Constructiong a torus (donut) from a filled rectangle.

3. Escher for the last time: Moebius band 2 (1963).

4.A Klein bottle.

5. A Klein bottle (movie).

6. Kleing bottle again (applet).

7. The projective plane (movie).

8. The projective plane (applet)

9. Travel in a three dimensional space. (movie).

10. Torus knots (script)