This seems to be turning into Mathematics week at peterjamesthomas.com. The “paradox” shown in the latter part of this article was presented to the author and some of his work colleagues at a recent seminar. It kept company with some well-know trompe l’œil such as:
However the final item presented was rather more worrying as it seemed to be less related to the human eye’s (or perhaps more accurately the human brain’s) ability to discern shape from minimal cues and more to do with mathematical fallacy. The person presenting these images (actually they were slightly different ones, I have simplified the problem) claimed that they themselves had no idea about the solution.
Consider the following two triangles:
The upper one has been decomposed into two smaller triangles – one red, one green – a blue rectangle and a series of purple squares.
These shapes have then been rearranged to form the lower triangle. But something is going wrong here. Where has the additional white square come from?
Without even making recourse to Gödel, surely this result stabs at the heart of Mathematics. What is going on?
After a bit of thought and going down at least one blind alley, I managed to work this one out (and thereby save Mathematics single-handedly). I’ll publish the solution in a later article. Until then, any suggestions are welcome.
For those who don’t want to think about this too much, the solution has now been posted here.