Archive for September 29th, 2008

The Double Möbius Star of David

September 29, 2008

Rosh Hashana begins today at sundown. It’s often referred to as the Jewish New Year, since it marks the start of the year 5769, but unlike the January celebration for the year 2008 it’s celebrated over two days. This is because Rosh Hashana always starts on the first day of the lunar month of Tishri, and traditionally months were officially declared by the Sanhedrin when a new moon was sighted. Messengers were sent out to share the date of the new month, but the most distant communities didn’t get the news in time and so to be safe they celebrated the start both 29 and 30 days after the previous month’s start. This custom continued even after the exact start of the month could be predicted mathematically.

As a math activity for these two days, you can make a Star of David out of interlocking Möbius strips (where “interlocking” is actually more like weaving). The half-twist in the Möbius strips means that the pieces will lie flat.

(I actually ran across this pattern on this jewelry site, which sells double Möbius stars in gold. Something to add to the collection of Möbius clothing!)

TwoPi observed (when looking over a draft of this post) that since in general it is possible to form interlocking links by cutting a twisted loop in half lengthwise, it might be possible to create this design by starting with a single loop and then cutting it. I tried, but wasn’t able to come up with anything: a loop has to have an odd number of half-twists to lie flat (like the Möbius strip itself), but if it has an odd number of half-twists then it will be a single twice-as-long loop when cut.

On the other hand, loops with an even number of half-twists form two interlocking loops when cut lengthwise, but the resulting loops will also have an even number of half-twists and so look a little bulgy. The closest I came was starting with a loop with two half-twists, but the resulting design just didn’t look as nice as the picture above.