Tuesday, September 27, 2011

simple origami and math: traditional envelope

Another very simple origami project that can help spark mathematical conversations is the traditional envelope. Like the other simple origami projects that I've mentioned in previous posts (the jumping frog and the paper cup), the instructions for the envelope are available from the Origami USA diagrams page.

This project is made with letter-sized paper (8.5 by 11 works great, I haven't tried A4), and is an accessible and appealing "practical" project (who doesn't like the idea of folding up a note so that it is its own envelope?).

The crease pattern (at the top of the post) is a great potential source of math-themed conversation. Identifying the types of shapes (the envelope itself is hexagonal - a rectangle with two corners cropped) and finding their area (what is the area of the final envelope compared to the size of the original note paper?) provide some things to explore. Parallel and perpendicular lines, and a few 45 degree angles, make talking about lines and angles in the pattern accessible for younger students.

The pattern is obviously symmetrical, but what kind of symmetry does it have? Many crease patterns that you might look at, like the paper cup (pattern below), have reflective symmetry. The envelope, on the other hand, has rotational symmetry (albiet a simple 180 degree rotational symmetry).

Something else to take note of is the "handedness" of the finished envelope. If you are careful when you follow the instructions, you will end up with an envelope with a front that has its top left and bottom right corners cropped (which is best if you want to affix a stamp to the top right corner).

However, if you are folding the envelope by watching someone else or folding from memory, you are just as likely to end up with its mirror image, an envelope that has its bottom left and top right corners cropped. The envelope, like a many modular origami units (like Sonobe units) has a right-handed and a left-handed version - if you fold a certain way you get one orientation, if you fold another way, you get the mirror-image. Which fold in constructing the envelope determines the orientation of the final model?

Postscript: Some more playing around with this origami model here.

Friday, September 23, 2011

punctured knight's tours

It's pretty clear that a chess knight cannot travel to every square on a 3x3 board. If the knight starts in the center square, it cannot move at all, while if it starts on any other square, it cannot reach the center. If you puncture the board by removing the central square, your dissapointment with the simplicity of the remaining problem might be somewhat relieved by the niceness of the solution: there is only one possible knight tour on a punctured 3 by 3 board (up to rotation, reflection, and change of direction), and it is closed with a nice star-shaped path.

This nice pattern inspired me to look at knight tours on other punctured square boards - boards with odd dimensions that had the central square removed. On boards that are 5 by 5 I could only find two distinct solutions (other non-distinct tours can be found by rotation, reflection, and reversing direction), but it is likely that there are more. Neither of the ones that I found are closed - the first follows a spiral path and always travelling in the same direction (like the 3 by 3 case), while the second starts out as a spiral in one direction and then changes direction after the ninth move.

However, I found that a nice closed tour can be created on a 7 by 7 punctured board by "gluing" together rotated copies of an open 3 by 4 tour. The technique of building up knight tours from smaller ones by gluing them together in a way that the knight can move from one to the next is a common one, and is particularly helpful when you want to create symmetric or semi-magic tours (this is described in Martin Gardner's essay "Knights of the Square Table" from Mathematical Magic Show).

Although this technique does not give you a spiral pattern like the 3 by 3 case or the first 5 by 5 example, the copy and rotate technique gives the path another nice pattern. You can see this symmetry in the 7 by 7 punctured board if you look at the cell values modulo 12 (doing this tracks where the values in the original board are rotated to).

If you connect the values that are equal to each other mod 12, you get a nice pattern of rotated nested squares - this pattern is completely independent of the tour on the initial 4 by 3 board: all it shows is the rotation that was applied to make the larger board. The image below has done this for some values (not all) - for example 5, 17, 29, and 41 form a square, as do 8, 20, 32, and 44.

This pattern is reminiscent of a more ideal version of the same pattern, which can be made using iterations in Geometer's Sketchpad (the gsp file used to create this is here). It seems that one way or another, we end up finding spirals.

You can use the same "rotate and glue" process to create a closed 9 by 9 punctured tour, made up of copies of a specially constructed open 4 by 5 tour. There are several open 4 by 5 tours that can be glued together to make a closed 9 by 9 punctured tour - here's one below:
Here are some previous posts about knight tours:
knight moves
closing time
kixote, or knight tour puzzles
more kixote

Saturday, September 10, 2011


More than any other book that I know of, Theodore Andrea Cook's The Curves of Life shows the extent of our fascination with spirals. First published in 1914, it is an odd blend of 19th-century natural history, amateur mathematics, and art history. On the mathematics of spirals, it is not the best source, Conway and Guy's The Book of Numbers has a better overview on spirals in plants, but it is unmatched as a compendium of all things spiral.

I was thinking about the allure of spirals while I finally got around to attempting some better renderings of spirals from earlier posts. The older pictures in this blog were made with Fathom, which worked well, but these drawn using Processing look a bit nicer I think, and the code is easier to play with.

The spiral below is a quadratic spiral displaying the triangular and hexagonal numbers, originally from this post.

This other spiral is a phyllotaxis spiral like the ones described here. The picture at the top of the post is based on the one below - with edges between points shown instead of the points themselves.

For what it's worth, the Processing code for these and other similar spirals is here. If Processing isn't your thing, you can find Mathematica and Python versions of polygonal-numbers-on-quadratic-spirals at Walking Randomly.