Amazing Moebius

This is a Moebius Strip in the creative work of M. C. Escher. A Moebius Strip has amazing properties. You can explore these by making your own Moebius Strip (without the ants!) quite easily. Take a strip of paper, give it a half-twist (turn one end over), and then tape the ends together. The result should look like the following:


Some things you might want to try are to put a dot in the center and trace a line along the "length" of the loop until you come back to where you started. Is this the same thing that would happen with a regular loop of paper? Now that you have a line along the center all the way around, grab a pair of scissors and cut along that line. Is this what would have happened with a regular loop of paper?

Much art, including sculpture by Max Bill and engravings by M.C. Escher have been based on the Moebius Stip and its properties. Stories have also been written, incluing "A Subway Named Moebius" by A.J. Deutch and "The No-sided Professor" by Martin Gardner.



And then there are the people who REALLY know how to have fun with a Moebius Strip, such as the creators of this video!

(Music by Didier Soyuz. Animation by Johnny Rem.)

Notice how the character goes around and around but is on top one time and bottom the other . . .

Tessellate

A tesselation is a tiling of the plane so that there are no gaps or overlaps - like a tiled countertop. M.C. Escher used tessellations in much of his work. Some were pure tessellations, like the first picture below, others used a tiling that changed into something else, like the second and third pictures below. The following images are his works (for more see the official M.C. Escher site).

Shodor has a great interactive website where you can make your own tessellations with the help of a Java Applet. I used it to make the tessellation at the top of this post. You can change colors and can choose to begin with a triangle, rectangle or hexagon and can alter these shapes using their sides or corners. I started with a hexagon. The computer helps you alter them in ways that will fit together. Have fun!

Where Do I Stand?!

Many optical illusions have mathematics at their foundation. Speaking of a foundation, where is the one in this image? Some optical illusions have their roots in the "impossible shapes" created by mathematicians Roger and L. S. Penrose, as can be seen in the work of M. C. Escher who frequently used these shapes as a basis for his art. Click here to find an optical illusions slide show.

Impossible Triangle & MCE

This object is known the Penrose Triangle or Penrose Tribar, after Sir Roger Penrose, the mathematical physicist (and recreational mathematician who created it. Dutch artist M. C. Escher was interested in mathematics and used this impossible mathematical shape twice over to create his lithograph "Waterfall" in 1961. Can you see where and how it is used? After figuring that out, you might want to check out the wild video on YouTube posted below Escher's work.

Mind Reader 2

Check out the Flash Mindreader Page and see if they really can read your mind!

(At the site, if you scroll down, you'll see they've provided a place for you to do your work so that you don't even have to do it in your head!! How's that for a deal?)

The image at left is a lithograph done in 1935 by artist M.C. Escher, much of whose work uses mathematical ideas.