Suspend disbelief, and is it possible that the earlier YGB surface is the connected sum of two separate Möbius loops? Here's one "failed attempt": https://0x0.st/zOWy.JPG https://0x0.st/zOW6.JPG Starting on yellow, if an ant is to take the high road, eventually it will pass two blue bars and return inverted to yellow. Then it may complete the Möbius cycle by traversing the high road again. Too bad, the ant will miss the two red bars on the low road. Similarly if the ant twice takes the low road it will miss the two blue bars entirely. Fortunately, there are four cycles (up to reversal), which pass both blue and red measures. So, if the ant chooses at random, most likely it will turn out to be a well-balanced ant (in the sense of having travelled red and blue... as well as green, yellow, brown and white). The homology count is just four choose two equals six, and all six cycles are measured by 4*Pi. This is enough to ensure the latest work different from the initial YGB. If the maker is more careful and does a "T" join on yellow, it should be possible to recreate a homology with exactly three Möbius cycles. I thought upon this construction the other day and doubted that the colouring could be preserved, but did not write out a rigorous proof. There is one final note on the Klein Bottle upcoming, but humoursly, we have run out of the teal blue thread, thus find ourselves in a lurch, waiting on a shipment from the merchant, ha ha ha. SPQR VOX VERITAS --Brad