Saturday, June 18, 2005

Conundrum

A few days ago while studying with my friend nSilico (who was deciphering a text on Graph Theory) he suddenly became agitated with an equation he was working with and stood up, shakenly proclaiming, "Oh my God! Math doesn't work!" I doubt any description can do justice to how fantastically comical this event was to me, but it was possibly the single funniest thing I've seen all year. I guess you just had to be there.

He later explained he'd spent hours learning how to employ a particular equation only to discover it never worked as its premises implied it should. He painstakingly had followed each element of the formula back to their original proofs only to reduce the entire problem to an obvious illogicality. It was as if the laws of mathematics had conspired against him and decided, today 1 + 1 = 3. As someone who'd devoted his life to studying physics, computer science, and mathematics, he was suddenly faced with a compromise of faith. I was witnessing a man losing his religion.

Eventually he divined the entire trouble was due to a typo he'd mistakenly accepted in the original text. A bit of fact checking confirmed this. After a brief bout of swearing and a letter to the publisher, all was well: He could continue to believe in the immutable truths inherent in mathematics.

Still, it's a scary and wonderful feeling to suddenly have all you know, or rather all you believe you know, challenged. I experienced a similar occurrence the very next day when looking a simple diagram of a triangle. Below I've provided an illustration so you can see what I'm talking about.




(use the slider to control the animation)

This puzzle bothered me to no end. I decided to animate it once I understood it. Apparently the laws of Euclidean geometry are no longer valid. How else can you explain the orange square left over in the end? (Warning - comments contain spoilers)

4 Comments:

Blogger nsilico said...

The apparent discrepancy in the area occurs because the trangles are not similar. The slope of the triangle hypotenuses are as follows:

1. Red = 3/8 = 0.375
2. Orange = 5/13 = 0.3846154
3. Blue= 2/5 = 0.4

So when the red triangle is located at the bottom left position, the aggregate figure has a "hypotenuse" that is slightly concave. When the red triangle is located at the top right, the aggregate figure has a "hypotenuse" that is slightly convex. This difference in the area under the "hypotenuse" between the two aggregate figures accounts for the remaining orange unit square in the final figure.

6/18/2005 5:33 PM  
Blogger nsilico said...

Although in the end the theorems of Euclidean geometry remain intact, it's still a cute problem. ;)

6/18/2005 5:36 PM  
Blogger Dædalux said...

Aww - you didn't even give anyone else a chance!

But wow - that was fast. Special figured it out quickly too. He Instant Messaged me rather than leave a comment. I imagine it originally took me a bit longer to solve it than the two of you, but we all figured it out the same way – by realizing the slopes were different.

Once you know that, you realize the overall triangle is an illusion. In my animation the orange ‘shadow’ actually has four sides. If it were a true triangle you'd notice its edge at the beginning of the animation when the ‘hypotenuse’ is concave. During the animation I shift one of its four points to match the new configuration, but this is (hopefully) masked by the movement of the other objects. The trick is - your eyes assume the whole thing is in fact a triangle.

I first found a static diagram of this figure on a website devoted to subliminal messages, but the illusion, known as Curry's Paradox, was originally invented by New York city amateur magician Paul Curry in 1953. It's a particularly good optical illusion because even when you know the answer your eyes remain fooled. It still looks like a triangle to me.

6/19/2005 12:15 AM  
Blogger Aanen said...

ugh I'm getting a headache from the math stuff, but the animation is pretty cool

6/20/2005 10:41 AM  

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