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Geometry Web Sites

> Does anyone have some good geometry websites?

* I recommend going to -> School Bell -> Math -> Geometry. There are lots of good links there.

* One of my favorites is not listed there: It is

Manipula Math, at
consisting of many Java applets which let you interact with geometric constructions.

* Another fantastic interactive geometry site (not listed in Yahooligans) is

Crystallographic Polyhedra at (click on "wireframe polyhedra," and also the link to the "crystal gallery," the "Java applets," or any of the other links). You can rotate the wireframe virtual 3D polyhedra by clicking and spinning them with your mouse.

* I also suggest you visit Math Cats, which is a new site (my site) recently listed in Yahooligans under Math Games and Puzzles, but it includes a math art gallery and a number of interactive geometry activities in the MicroWorlds section. Also, visit the new unit in the Explore section on how to make 3D polyhedra (tetrahedron, hexahedron, octahedron, decahedron, and icosahedron) consisting of linked equilateral triangles created with just a compass and a straight edge (and scissors and glue).

Wendy P. of Math Cats, 10/19/00 on math board

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math and art sites

> I am looking for the some art ideas that incorporate math.

There are three sections of Math Cats which feature an integration of math and art:

1) the Math Cats' Art Gallery, which displays a large slideshow of geometric art created by Logo users from around the world with 15 words or fewer of Logo programming code. The codes are provided on an accompanying page, and there are links for downloading free versions of Logo for creating such artwork yourself. By the way, next to each author's name on the page showing the code is that person's home page, when available. Many of these creative people have created a lot of geometric art of their own, so you should follow some of those links, too (especially Yehuda Katz of Israel and Olga Tuzova of Russia... also Keith Enevoldsen, listed below).

2) the Math Cats Explore section, which currently features step-by-step instructions for making geometric space forms with a compass and straight edge. More forms are coming soon.

Also in the Explore section is the Lissajous Lab, which allows the user to create an endless number of beautiful animated string art designs by manipulating numbers.

Another project which will be added to the explore section within the next few days [it is now posted] will be Polygon Playground, which will allow users to arrange dozens of polygon outlines of different sizes and shapes to form symmetrical designs, tessellations, abstractions of familiar things (such as people, animals, etc.)

[The Math Cats Explore section now features more activities integrating math and art: String Art, Tessellation Town, Encyclogram, Exploding Math Art, and illustrated Number Stories.]

3) the Math Cats MicroWorlds section, which has numerous interactive projects for creating geometric art, including: symmetry, tessellations, "spiderwebs" (string art), Olympic rings, and more.

Other favorite math/art sites:

One of the most intriguing math/art sites I've found is:
which features images of polyhedra which the user can rotate to examine from any angle.

Another favorite math/art site is:
which features photos of gorgeious handmade geometric sculptures, over 1,000 virtual polyhedra, links to other sites for geometric art, and more.

And please visit a little-known site, the home page of Keith Enevoldsen, a quietly-creative person who initiated the 1997 15-word Logo contest which is now exhibited, with his permission, in the Math Cats art gallery. He has several fascinating math/art projects on his site, one of which displays words with rotational symmetry; another combines geometric designs to form animations. (I also love his "gibberish generator," which is unrelated to math but extremely entertaining! His site is:
(and according to his site counter when I visited just now, there have been only 205 hits to his home page... and a number of those were from me!) also includes some links to math art sites (fractals) and has a display of geometric space forms built by the site's creator, Karen.

Wendy P of Math Cats, 11/16/00 on math board

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> Hi, I am looking for some different games, art projects,
> centres to use for our geometry unit. (We learn the basic
> prisms and faces, edges, points etc)
> Karen, 5/31/00

Try: (do a geometry search) (do a geometry search)

Roberta, 5/31/00 on math board

Geometry projects for at-risk students

> I need some project ideas for my at-risk students in
> Geometry. This, of course, is Informal Geometry. I
> believe in "hands on" learning for these students, but I'm
> out of ideas. My experience is limited to teaching
> Geometry with proofs to college-prep students.
> Janet, 9/23/00


The best all-encompasing project I've used is called "Polyhedraville." (What I can't tell you is where to purchase the book that guides you through the project.) I did a search on the web, but found only descriptions of what it is, and some pictures of completed projects. Maybe this will give you a start toward investigating it.

Sylvia/CA, 9/23/00 on math board

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My favorite part of teaching geometry is the constructions. You can use "safe" compasses or patty-papers and folding. When you get up to teaching the lines in a triangle (altitudes, medians, angle bisectors), teach them how to CONSTRUCT these, after they learn to define them and draw them. I have a project where I ask them to draw four scalene triangles on separate sheets of paper (you could draw it for them and then xerox and hand out the necessary copies). On the first one, they must CONSTRUCT the three altitudes , on the second triangle, they construct the medians, then on the third, the angle bisectors. On the fourth they will construct the perpendicular bisectors for each side.

The next task is to cut out one of these triangles from stiff poster board or cardboard and construct all of these lines (each TYPE of lines should be a different color) and note the points of intersection of each type. They need to discover which of the intersection points will be the center of the circle circumscribed around the triangle (the orthocenter), which intersection point will be the center of a circle inscribed in the triangle (the incenter), and which intersection point will be the center of gravity of the triangle (the centroid). (That is why they need to cut it out - the triangle should balance on the point of a pencil if they balance it on the correct point.)

They need to hand in each sheet with construction lines displayed, and a sheet explaining which is the incenter, orthocenter, centroid.

Easier projects can involve:

mobius strips,
string art,
parabolic arch posters (very creative and artistic),
flexagons (see Martin Gardner's Scientific American books -- should be available in libraries),
tessellations (check out,
optical illusions (Mark Newbold's animated necker cube pages -- -- include explanations of why the illusions work and many are geometric in nature),
and let us not forget symmetry using snowflakes and spirograph-type designs.

Just spread these topics out throughout your curriculum wherever they fit in as most relevant.

Oh, and from a textbook I once used, have them fold patty paper in different ways and then punch holes through the thicknesses in the most folded corner. In some sort of simple pattern - an arc or a straight line. Without unfolding, try to visualize what the pattern of the holes will look like when the paper is unfolded. Just start with one hole, then two...

DSF/NJ (Donna), 9/23/00 on math board

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