Steve's 4D web pages.
These pages are for the Four Dimensions short term class.
4D web -- this page shows work by former Paideia students and it just interesting in general. Here is the link to the 4D what to do page .
Topic Try this Due Grab Drawing helps you see. It does not matter if you draw well, the visual benefit is the same. Go to this web page ; read page 1, draw the figures on page 1. Second try to draw the 4 figures on page 3. You need not try the Escher one at the bottom. These pages were created by Thomas Leavett who graduated several years ago.
Type "polyhedra" into Google. Choose a site and explore the various polyhedra. Try to find things that are interesting. Remember -- simpler is better. You can build sophisticated things from simple thoughts. Complicated things are just complicated.
This is the web pages in pdf form. You do not need this as it will all be on the web.
On the internet (perhaps at a site you visited last night) look up "cubeoctahedron." Understand what it it when you come to class.
Read the last two pages of this web page (you went here yesterday) written by Michael Matthews.
Visualization problem1: in the plane (ie., on a sheet of paper), put 6 points so that three of the points are the same distance apart; two of them are the same distance apart (not the same as the first three though), and one distance is unique.
Visualization problem2: a cube has a long diagonal that connects a vertex to its opposite. Rotate the cube rapidly around this axis, as explained in class; draw a picture of its shape. This one is harder.
From these web pages print out the pictures. Cut out each image and fold along dotted lines. Tape the figure together. Each will become a vertex of a polyhedron. We will assemble them into polyhedra in class. The purpose is to set up the classical proof that there are only 5 regular convex polyhedra.
Here are some pictures drawn by former Paideia students Emily Englehardt and Michael Matthews. What are these pictures trying to show?
If you have some dice at home, collect as many different types as you have. Read Thomas' description of symmetry and, for each die, try to determine the symmetry of each edge, each vertex, each face. This is to prepare for the group properties of polyhedra.
Visit some of your 3D sites.
Type the word "polytope" into Google, find a site that allows you to rotate them. Just look and see what you can see.
Using scissors, paper, and tape, do the "cutting and pasting exercises" on the handout from class.
Check out David Emory's Topological folding pages given out in class. Cut out a rectangle and try some of them yourself. Hint: they are not all possible in 3D space; trying them will help you see why.
Visualization problem: The front view of an object is a sphere; the side view is a sphere; there are no hidden lines. Draw the object or draw its top view.
The handout you got in class is Paul Smith solving the last visualization problem. Remember, visual stuff can be difficult but the attempt to visualize is always beneficial here, where you will find a hypercube analyzed into cells, each of which has its dual octahedron. Very good for visualization. Try drawing a hypercube and putting a point in the middle of each face of a single cell, and connect them to form that cell's octahedron. Do this for several cells and bring it to class.
there, where you will find a lot of pictures, especially the last 5. Try to figure out what the last 5 are and be prepared to talk about them in class.
Cutting and pasting; tape or paste two cylinders together as shown in class, intersecting perpendicularly in one place. Cut down the midlines. As you cut, you want the figure to come apart at the cuts so the rest must be taped or pasted securely. Second time, same figure and procedure but one has a Mobius twist.
For Tues., Jan. 11 Read Michael Matthew's description of duality. This is do good you might want to print it out.
Check out Michael and David Akers' work done at Brown University; Focus on duality and the slicing applets.
new web stuff.
Justin Baker's 4D (some math; lots about how we see 4D)
Again check out Michael and David Akers' work done at Brown University; Find things that you did not read yesterday.