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Monday
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Wednesday
- Read: Handout on Growth and Symmetry, from Tannenbaum & Arnold.
- Homework (due Friday, February 6):
- Suppose you start with a golden rectangle (so the ratio of the
long side to the short side is equal to "phi"). Use algebra to show
that the following procedures will produce more golden rectangles:
- fold the original golden rectangle in half vertically and then
horizontally
- add a square to one side of the original golden rectangle
- Exercises 45, 51, and 61 from Tannenbaum & Arnold.
- Discussion Questions for Friday's
class discussion about the Golden Ratio and Golden Rectangle.
- Event: Tea & Games, 2-4pm in the Swords Student Lounge. Drop by
for some hot chocolate and a game of Abalone, or Set, or Rush Hour
(yes, we have Rush Hour again!)...
- Some Web Links:
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Friday
- Study: for Quiz 1, which will be given at the beginning of class
on Monday. Expect one question on polygons (like the HW and our class
discussion from Friday, Jan 23), and one question on
algebra (solving an equation, simplifying, etc.)
- Read: Continue reading Flatland. Plan to finish it before
next Wednesday, February 4th.
- Writing: Assignment 2, due Friday,
February 6 at the beginning of class.
- Solutions to Assignment 1,p1,p2,p3,p4,p5,p6
- Some Web Links:
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Monday
- Study: chapter from Tannenbaum & Arnold, and class notes from
today on history of Fibonacci numbers and their occurrence in
nature.
- Assignment: Nothing new today.
- Some Web Links:
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Wednesday
- Study: Today's group problems with the Fibonacci numbers and
Lucas numbers. We will continue this activity in class on Friday.
Your solutions to this activity will be due Wednesday February
11th, as part of our next weekly homework assignment.
- Some Web Links:
- The Fibonacci
Association. The Fibonacci numbers attract a lot of attention!
This organization publishes scholarly journals, holds conferences, and
has a wealth of resources on the Fibonacci numbers.
- Quiz 1: Has been graded and may be picked up tomorrow during
office hours. I will bring the rest to class on Friday.
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Friday
- CLASS CANCELLED.
- Read: Finish Flatland. We will start discussing Flatland and the
concept of dimension on Monday.
- Fibonacci Group Problems: Will be due next Friday. I'll find an
evening time when we can discuss these further. More on this later.
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Monday
- Study: The arguments about "cross sections" and "terminal points
and sides" in Flatland. We will discuss these in class on Wednesday,
in addition to some other facts regarding dimension.
Also begin
studying for Quiz 2 which will be given at the beginning of
class on Friday. Be prepared to answer basic questions about the
Fibonacci and Lucas numbers (i.e. give their definitions, be able to
compute them, explain what notation like F_N, F_{N+2} means, etc.).
Also be prepared to answer a question involving the Golden Ratio and
geometry. This would be similar to our work in class with golden
rectangles, and/or to your homework involving golden rectangles and
golden triangles.
- Read: The handout Flatland: A New Introduction by
T. Banchoff, and Chapter 3 of Peterson (A Place in Space).
- Homework (due Friday, February 13):
- The group problems on
Fibonacci and Lucas numbers that we began discussing in class last
Wednesday.
The last question is optional. (Bonus points will be given on this
one!) Also, I will be available Wednesday, 6-8 pm in our usual
classroom to meet with anyone who would like to work on these problems
with help available. You may also ask about them during office hours.
- Some Web Links:
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Wednesday
- Study: for Quiz 2, which will be given on Friday, at the beginning
of class. (See Monday's entry for details.)
- Read: Handout from Kinsey & Moore, from Ch. 6 "Other Worlds, Other
Dimensions" and Ch. 13 "Topology."
- Writing: Assignment 3 on Flatland, due Friday,
February 20 at the beginning of class.
- Homework (due Wednesday, February 18):
- As we discussed in class today, regardless of how a sphere passes
through Flatland, the cross sections will always be circles. Take a
solid cone and show how two different orientations would give rise to
two different sequences of cross sections. Please illustrate your
solution as needed.
- Think about what a circle looks like as it passes through
Lineland, and what a sphere looks like as it passes through Flatland.
By analogy, what would a 4-dimensional sphere look like as it passes
through 3-dimensional space?
- On pages 60-61, the sphere discusses how to construct a square by
moving a line parallel to itself. Suppose instead that the sphere
issues these instructions:
Start with a point. Connect
that point to another point. Moving in a new dimensional direction,
create a new point and connect it to the previous two. Moving in a
new dimensional direction, create a new point and connect it to the
previous three. Continue this process. As much as
possible, draw the figures that would be constructed and for each one,
determine the number of vertices, edges, and faces.
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Friday
- Study: our notions of covering dimension and embedding dimension
from today's class, and pp. 411-420 from the most recent handout from
Kinsey & Moore.
- Read: brush up on the 4th dimension and "hypercubes" in Chapter 3
of Peterson.
- Homework: none posted today
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Monday
- Study: Covering dimension and embedding dimension. The
vertices/edges/faces/solids constructions from today's class.
- Read: Today's handout "Cubism and the New Geometries" from The
Fourth Dimension and Non-Euclidean Geometry in Modern Art, by
L. Henderson.
- Homework (due Wednesday, February 25):
- Determine the dimension of the following objects, using
each of the three methods we discussed in class: (1) counting
directions, (2) exponentiation, and (3) covering dimension. (You
should get the same answer from each method, so the exercise is to understand
and explain why.)
- the surface of a sphere (it will help to know that the
surface area of a sphere is 4*pi*r^2 where r is the radius).
- the perimeter of a square (in terms of its side length s).
- a collection of points.
- Determine the embedding dimension of each of the objects
in the previous problem. Explain your answers.
- Draw the unfolded cube from today's class, and give instructions
about how to glue the sides together to form the cube. (e.g.
label sides to be glued to each other with a common symbol or letter.)
- In the view of a hypercube shown on p. 50 of Peterson, visualize
and count the number of vertices, edges, faces, and solids. (You have
the numbers from today's class, so your job is to verify them.)
Indicate/label these vertices/edges/faces/solids on a copy or copies of the
drawing.
- EXTRA CREDIT: Draw or build the unfolded hypercube
from today's class (made out of 8 blocks, shown on p. 44 of Peterson).
Give instructions about how to glue the faces together to form the
hypercube!
- Some web links:
- Spirits,
Art and the Fourth Dimension, an interesting article at
strangehorizons.com. Towards the end of the article, there are
excellent images of Duchamp's "Nude Descending a Staircase No. 2,"
some cubist portraits of Picasso and Metzinger, and Dali's "The
Crucifixion."
-
Cut The
Knot! An interactive column using Java applets (including an
animation of today's constructions of the line/square/cube/hypercube),
and a lot more cool links.
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Wednesday
- Study: our work on constructions (square, cube, hypercube) and
cross-sections. We will have a quiz on this material next Monday.
- Read: Ch. 6 of Peterson, "Crystal Visions."
- Homework (due Wednesday February 25):
For these problems, you
may use the following digital
color models. Please give as much numerical description as
possible in your answers.
- Find a way to code up the four squares in the upper right hand
corner of the Ellsworth Kelly painting we looked at in class. For each
square, give a location in the xy plane and decide on color values
that approximate the color of the square. (You can use the RGB model,
or the HSL model.)
- Using the RGB model, describe:
- two different paths from black to white that never use
blue. Tues Feb 24: What this means is a path from black to white
where as the color changes, it will never appear as blue. (You will
need to use a B-value to get white eventually, but you want to change
this value last.)
- a path from black to white in which the colors traced are always on a grayscale.
- Using the HSL model, describe:
- a path from blue to red
- path from black to white that passes through red
- a path from black to white that stays within a grayscale
- Some web links:
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Friday
- Entry corrected on Monday 2/23
- Study: Our work on covering & embedding dimentions, constructions
(square, cube, hypercube) and cross-sections. This material will be
quizzed next Wednesday.
- Read: Chapter 11 on Symmetry, in handout from Tannenbaum &
Arnold.
- Homework: nothing new for today.
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Monday
- Study:
- For Wednesday's quiz: covering and embedding dimensions,
cross-sections of an object when it passes through a space with lower
dimension, and our geometric constructions of the line, square, cube,
hypercube (including # of vertices, edges, faces, etc.).
- The Tannenbaum & Arnold handout on Symmetry. On Wednesday after
our quiz, we will start discussing the patterns which repeat in
two directions, known as the Wallpaper Patterns.
- Homework (due Wednesday March 3):
- Today's in-class activity on classifying the symmetry type of
border patterns. For each pattern in the packet, determine its
symmetry type using the flowchart, and enter your results in the
table. With some pictures, there may be more than one element that
can be considered as a border pattern. If so, treat each one as a
separate example. Also note that with reflections, one has to be
careful. If any part of the motif is curved, what appears to be a
reflection may not actually be one!
- Some web links:
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Wednesday
- Read: Handout from Kinsey & Moore on Two-Dimensional Symmetry, and
the "wallpaper patterns".
- Homework: nothing new today
- Some web links:
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Friday
- Study: Class notes and today's group activity on the 17
Crystallographic Patterns (a.k.a. "wallpaper patterns.").
- Read: (handout) Chapter 1 from Visions of Symmetry, about
the artist M.C. Escher.
- Homework (due Wednesday March 3):
- Today's in-class activity on classifying the symmetry type of
the wallpaper patterns. For each pattern in the packet, determine its
symmetry type using the flowchart, and enter your results in the
table. As noted in the instructions, you should indicate (1) the
"generating region," (2) the symmetry type, and (3) the grid pattern
used in constructing each example. Also, for the 3 types that Escher
did not draw, you will need to draw your own example. Please have fun
with this! (There are examples in the Kinsey & Moore reading, for
your reference.)
- Some web links:
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Monday
- Study: rigid motions, symmetries in Frieze patterns, and
symmetries in Wallpaper patterns, for Wednesday's quiz.
- Read: Ch 1 on Penrose Tilings (handout), from Gardner's book
Penrose Tiles to Trapdoor Ciphers.
- Homework (due on Friday, March 19):
These questions are all
about Penrose's kites and darts which were demonstrated in
class. In each question you should make the examples from the
kites and darts that were handed out in class. Once you have your results,
tape them together & to the page, to turn in.
- We saw two of the ways that kites and darts can be placed around a
vertex, using the matching rules:
(1) put 5 darts together at the
vertex, which gives the "star" pattern, and (2) put 5 kites together
at the vertex, which gives the "sun" pattern. Another possibility is
to put together the heads of three darts, and then fit in the wings of
two kites. This makes a figure known as the "king." There are only four
more ways to put tiles around a vertex. Find them!
- If there is only one choice for fitting a tile around a group of
vertices, we said that the tile was in the empire of the group
of vertices. We saw that there were at least 10 tiles in the empire
of the star, since we were forces to have one "round" of 10 kites
surrounding the star. Prove that there are exactly 10 tiles in the
empire of the star, by finding two different examples of how you could
tile the next "round".
- EXTRA CREDIT QUESTION: Find the empires for the four vertex
groupings you found in Problem 1. (HINT: You should find that one of
them has no tiles in its empire, one has two tiles, one has 11 tiles,
and one has 22 tiles.)
- (One you should all do.) Put together a tiling, starting from the
vertex grouping of your choice, that uses at least 4 "rounds" out from
the center. (Try to come up with something different than the
examples in the handout!)
- Some web links:
- Tesselations,
at mathartfun.com, a site where you can buy cool puzzles, posters, and
books on tilings of the plane.
- A nice site on aperiodic
tilings, including Penrose tilings. (This site includes some great
illustrations, including our example of a tiling using Penrose's original set of
6 tiles.)
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Wednesday
- Study: Instructions for how to fold the PHIZZ origami units and
the Sonobe origami units.
- Read: Cipra article on origami (handout from today's class), and Chapter 4 on "Plane
Folds" in Peterson.
- Homework (for class this FRIDAY):
- Fold 30 of Tom Hull's PHIZZ units, using the instructions from the
handout and the small white, yellow, and blue paper squares.
- Fold 6 of the Sonobe units, using the instructions from the
handout and the larger green, purple, and cream paper squares.
- Bring all of this to class on Friday!
- Some web links:
- Joseph Wu's origami
page, with a gallery of images from his collection and from origami
conferences. Also included are folding instructions for many models.
- Tom Hull's
homepage, with lots of origami-related links, including a nice gallery
of images.
- Meenakshi Mukhopadhyay's Modular
Mania page, with images and instructions for a plethora of
models!
- Helena
Verrill's origami pages, featuring info/images on modular origami
and tesselations.
- Andy Wilson's Tesselations
page.
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Friday
- Study: Chapter 4, "Plane Folds" from Peterson.
- Read: Handout on graph colorings from today's class.
- Writing Assignment 4: due Friday, March
19th. This assignment is the first stage of the course project. Be
sure to read all instructions carefully, and ask me any
questions you might have about the format/expectations.
- Homework (due Friday, March 19th): Origami!
- With your 30 PHIZZ units, build your own copy of the model of the
dodecahedron that I brought to class. Full credit will be given for
constructing one, and EXTRA CREDIT will be given if your model is
properly 3-edge colored. If you weren't in class and need
instruction about how to put the units together, stop by office hours
sometime and I will be happy to help.
- With your 6 Sonobe units, build your own cube (again, like the
model I brought to class). You have two units each of three different
colors, so you should properly 3-edge color your cube. (This time
it's not extra credit, because it's a much simpler exercise!)
- EXTRA CREDIT: Fold me something interesting -- your favorite
origami model!
- Some web links: Nothing new this time. (Check out the links from
Wednesday, if you haven't already.)
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Monday
- Study: Today's derivation of the formula for fractal
dimension.
- Read: Chapter 12 of Tannenbaum & Arnold: "Fractally Speaking,"
and
Chapter 5 of Peterson: "Grid Fields."
- Homework: None today.
- Some web links:
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Wednesday
- Study: Our work on generating fractals and computing fractal
dimension, from last class, and from today's group activity.
- Read: nothing new today.
- Homework: nothing new today. Remember that the HW on kites and
darts, and on origami, is due this Friday.
- Some web links: nothing new today.
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Friday
- Study: Our work on generating fractals and computing fractal
dimension, from last class, and from today's group activity.
- Read: nothing new today.
- Homework (due Monday, March 29th at the end of the day): Due
date corrected on Friday 3/26
The
problems in today's activity on Fractals and Dimension, continued from
Wednesday.
- Reminders/Announcements:
- As previously announced, there will be NO CLASS next Monday
and Wednesday, March 22nd and 24th. (Prof. Frechette will be away at
a research conference.) These two classes will be made up during the
last two weeks of classes during the project presentations, at a
time and location TBA.
- Our next quiz will be on Wednesday, March 31st, on fractals. More
information on what kinds of questions to expect will be given next
Friday in class.
- Some web links:
- The Dynamical Systems and
Technology Project at Boston University. A tremendous amount of
information about fractals, and some great JAVA applets for creating
them.
- Cynthia Lanius'
Fractals page. Starts with an answer to the question "What's so hot
about fractals, anyway?" and has great examples and demos you can do
online.
- Fractal
Resources at the Fignations of Imaginment site. This includes a
huge list of fractal software, much of it freeware or
shareware, including some software for making fractal music!
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Monday
- Study: for Wednesday's quiz on fractals. You should be prepared to do the
following:
- Draw the 2nd iteration of a fractal, given the base shape and the
replacement rule.
- Compute the dimension of a fractal, given the base shape and the
replacement rule.
- Explain what "self-similar" means.
- Homework (due Friday, April 2nd): The second component of our
course project: AN OUTLINE. Be sure to re-read the instructions
carefully before preparing and submitting your outlines!
- Some web links: see entry from Friday, March 26th.
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Wednesday
- Read: today's handouts on finding fractals in art.
- Homework (due Wednesday, April 7th):
- Try out some of the Explorations in the Mandelbrot Set
Explorer, which we used on Monday and last Friday.
- In particular, check out the Mandelbrot
Set Iterator applet. Here's something we didn't try: zooming in. If you
left-click on the picture and drag the mouse, it puts a frame on the
picture. If you then clidk on the "compute" button, the window will
zoom to that frame. Also, if you change the number of iterations and
then hit enter, the picture will be computed in finer detail. Try
playing with this, "zooming in" repeatedly on sections along the
fringes of the Mandelbrot set.
- For each of the given values of c:
c = -2, c=-2i, c=-1/4 + 3/4 i, and c=3,
do the following:
- Compute the first 7 terms of the
corresponding Mandelbrot sequence M_c, as defined in
class.
- Based on the trend in the numbers, determine whether or not c is
in the Mandelbrot set. (Recall from class that the criteria for
membership in the Mandelbrot set is that M_c be either periodic or
attracted.)
- Verify your answers by going into the Mandelbrot
Set Iterator applet and finding the location of c in the picture.
- Some web links:
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Friday
- Read: nothing new today
- Homework: nothing new today. See Wednesday 3/31's entry for homework on the
Mandelbrot set, due on Wednesday 4/7.
- Some web links: Pictures from today's window-taping activity can be viewed
below:
- Photo 1, Photo 2, Photo 3, Photo 4, Photo 5, Photo 6, Photo 7, Photo 8, Photo 9, Photo 10, Photo 11, Photo 12, Photo 13, Photo 14, Photo 15, Photo 16, Photo 17, .
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Monday
- Study: Lessons 1-4 of Frantz, and our class notes on perspective.
We will have a quiz on this material on Monday, April 26th. More
details to come in class on Wednesday.
- Read: Edgerton article The Heritage of Giotto's
Geometry... (handed out in a previous class), and
today's handout from Baxandall, Painting and Experience in
Fifteenth-Century Italy. We will have a discussion on these
articles in class on Wednesday.
- Homework: (due Monday, April 26th)
- For each of the three tilings handed out in class today, transfer
a portion of them onto the floor in a one-point perspective drawing.
Your final drawing should show enough of the tiling so you can see
some repetition of the pattern.
- Reminder: Course projects are now due on Friday, April
26th. Please remember that both your pair's paper and your
original artwork are due on that date. I will collect a copy of each
(or a copy of the paper and a photograph of the artwork, as needed).
Presentations have been scheduled and will begin on Friday. Please
check the schedule handed out in class, if you need to recall the time-slot for which you
signed up. Also, if you have any questions about the presentation and
would like to discuss them with me beforehand, please stop by office
hours or make an appointment.
- Some web links:
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Wednesday
- REMINDER: Course projects are due on Friday, in class.
Please turn in a copy of your pair's paper, along with a photocopy or
photograph of your original artwork.
- Read: Handouts from Edgerton's The Heritage of Giotto's
Geometry: Art and Science on the Eve of the Scientific Revolution
and Baxandall's Painting and Experience in Fifteenth-Century Italy.
- Writing Assignment: (due Friday, April 30th)
Write a reflection paper about these readings, focusing on the questions
from today's handout. The two main themes these questions touch upon
are
- The mathematical skills of the average person in 15th century
Italy, and how these contributed to both the development of the method
of perspective and the reception the method received from the
public.
- The people's desire to understand and perceive the
world according to God's vision.
Your assignment should be a minimum of one page in length.
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Friday
- Study: For Monday's quiz on perspective. In particular, as stated
in class on Wednesday, you should be prepared to do any of the
following:
- Find the principal vanishing point in a given picture, by
extending lines which represent parallel lines in the real world.
- Compute the coordinates x' and y' for the image
point P'= (x', y', 0) in the picture plane, given a point P = (x,y,z)
on the object being drawn/painted.
- Fill in missing lines in a picture drawn in 1-point perspective,
by making use of the vanishing point, and of your knowledge of how to
find the center of a rectangle, etc.
- Some web links:
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