The triangle paradox


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:

Old or young woman?

and

Quadruped?

and

Parallel lines?

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:

Spot the difference...

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.
 

7 Responses to The triangle paradox

  1. Certainly not a mathematician myself, I can see how the red triangle covers slightly more space than the green on, resulting in a crooked line (when I tilt my mobile that’s easy to see)
    Likewise, the first picture has a hollow line, when looked at from the side

    So basically, this picture is a fake. But I can imagine rigid mathematicians getting all excited over it

    • Peter Thomas says:

      Hi Martijn,

      Thanks for the comment – as a Mathematician myself I’ll try not to get too offended about the rigid comment. We tend to prefer the related word, rigorous.

      Wasn’t too sure about what you meant about a hollow line, but I think you may have got the essence of the issue in your previous comments.

      BTW the picture is not a fake, it is to scale and accurate (subject to shifting pixels when resizing and the accuracy of Visio that is). The issue is what we see rather than what is there – same as the first three images.

      Peter

  2. Ajay Ohri says:

    No such thing as a straight line in the picture, given x/y intersection of diagnol.
    Interestingly the effect of optical illusions in data visualization or business intelligence reports is something to be pondered on.

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