the Spider's Gedankenexperiment / Pride Goeth Before a Fall

Clicking the image is a great idea.

RamanuJohn said...

You don't need imagination for this -- you need the math!

Let's say the spider needs to cross the 20-unit edge, either at a corner or somewhere along the edge. So the spider's distance is the sum of two Pythagorean formulas, each containing "x" to account for the place where the spider traverses the long edge. Find the derivative and solve for 0. "X" equals 10, meaning the spider crosses the 20-unit edge right in the middle. Total distance: 30.54 units. A quick check of any path that traverses the 10- or 13-unit edges shows that any such path is longer than 30.54 units.

Tuesday, 11 March, 2008

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"Pride goeth before destruction,
a haughty spirit before a fall."

לפני-שבר גאון, ולפני כשלון גבה רוח

-- Proverbs 16:18

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I guess you could call the Trick what the old Euro physicists called a Gedankenexperiment, a Thought Experiment.

In Real Life, you can't really put hinges on a living room and then fold back the ceiling to make a single plane from corner to opposite corner.

But the Spider can do it in its imagination, and that leads to one simple right-triangle straight-line geodesic that is indeed the shortest crawling path the Spider could take.

Back in the Real World of Real Living Rooms: To the Spider, it will seem like two different straight lines. But they sum to 30.48 feet, well less than 30.54 feet.

I suppose that puts a burden on the Spider of figuring out its starting-out angle. But that's simple trig, which I am told Spiders are very good at.

When it reaches the ceiling, things get easier, it just has to aim straight for the final corner.

You can put the hinges somewhere else and fold a wall back to make a different plane, but that solution path is longer.