Geometry

The Perfect Parallelepiped

In general, it’s unusual for a rectangle to have sides and diagonals whose lengths are all integers (i.e., whole numbers). Consider the following three rectangles, all of width 3:

Looking at the different lengths, we see one place where the diagonal length is an integer, but in the other cases, the diagonal length happens to be a non-terminating, non-repeating decimal (i.e., irrational). Indeed, the diagonal length will be an integer exactly when the length and width are part of a Pythagorean triple, but compared to the alternative, this is uncommon. (While there are infinitely many occurrences of this, we can still meaningfully consider it uncommon).

Now, imagine the situation in three dimensions. A rectangular prism (think of a cardboard box) has 12 sides, 12 face diagonals, and four space diagonals. It would be extremely unusual for all of those 28 lengths to be integers. Even if we didn’t limit ourselves to rectangular prisms, but we allowed for the box to be slanted in all directions (that is, a parallelepiped), it would still be a numerical miracle for all those lengths to be integers.

Well, meet the perfect parallelepiped!

This was discovered by a couple of mathematicians at Lafayette College in Pennsylvania, using brute-force computer trials. It looks like they found some others, too. So thank you, Clifford Reiter and Jorge Sawyer, for giving me an extra credit problem for my next exam!

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CDs, Prisms, and Parallelepipeds

By patrick honner, 14 years8 years ago

The Final Word in Triangle Appreciation

As has been previously noted, October has been a nice month for triangle appreciation. Today, 10/19/10, offers us one more polygon to ponder.

Consider the 10-19-10 triangle.

It’s a fairly ordinary triangle, as triangles go. It’s a little short compared to the other triangles we’ve looked at recently, but there’s nothing wrong with that.

What’s special about the 10-19-10 triangle is that it’s our last chance this month to enjoy triangularity. Tomorrow, the Triangle Inequality steps in. You can’t have a 10-20-10 triangle, because once the third side hits 20, you need all of the 10s to get from point A to point B. There’s no wiggle room for the triangle’s interior.

Although I must admit, I do find much to admire in this degenerate triangle.

Geometry History

Benoit Mandelbrot

Benoit Mandelbrot died this week at the age of 85, leaving a giant mark on the worlds of mathematics and science. Mandelbrot coined the term fractal, writing the seminal book on the topic in 1982–The Fractal Geometry of Nature. By rejecting the generally accepted, if never actually articulated, notion that things were smooth, Mandelbrot challenged everyone’s notion of shape, distance, and dimension.

As is often the case, Mandelbrot was considered crazy at first, but he dies with almost legendary status. We will likely be talking about fractals and Mandelbrot sets hundreds if not thousands of years from now, the way we talk about the Pythagorean Theorem and Euler’s Number today.

And as is also often the case, Mandelbrot’s brilliant and revolutionary ideas can be traced to a simple question that he chose to think of differently: how long is the coast of Britain?

The answer, he was surprised to discover, depends on how closely one looks. On a map an island may appear smooth, but zooming in will reveal jagged edges that add up to a longer coast. Zooming in further will reveal even more coastline.

“Here is a question, a staple of grade-school geometry that, if you think about it, is impossible,” Dr. Mandelbrot told The New York Times earlier this year in an interview. “The length of the coastline, in a sense, is infinite.”

From Mandelbrot’s NYT’s obituary.

By patrick honner, 14 years9 years ago

Appreciation Geometry

More Triangle Appreciation

It’s been a great week for special triangles.

Six days ago was a rare Equilateral Triangle day, four days ago we appreciated the 10-12-10 triangle, and today, 10/16/10, gives us another triangle to admire.

It’s actually very closely related to the 10-12-10 triangle, which we saw is just two right triangles pasted together.

Just cut along the dotted altitude:

Now rotate the two pieces:

Stick them back together along their common side of length 6:

Now flip, and voila! Another triangle made by gluing two congruent right triangles together!

I’m running out of good triangles, so we may not appreciate another until December 10th.

Geometry

Today in Triangle Appreciation

As it’s 10/12/10, I thought it would be the appropriate day to appreciate the 10-10-12 triangle!

Of course, it’s nice that the 10-10-12 triangle is isosceles. But what’s really cool is what happens when you drop the altitude from the top vertex!

It’s well known that in an isosceles triangle, the median and the altitude from the vertex are the same–this means that not only does this segment make a right angle (it’s the altitude), but it also divides the opposite side at its midpoint (it’s the median). So that segment creates two right triangles with hypotenuse 10 and side 6. Of course, the other side must be eight, since

$6^2 + 8^2 = 10^2$

Thank you, Pythagorean Theorem. So the 10-10-12 triangle is just two right triangles pasted together.

It’s not quite equilateral, but the 10-10-12 triangle is still pretty cool.

Geometry

Geometry Technology

The Perfect Parallelepiped

Geometry

The Final Word in Triangle Appreciation

Geometry History

Benoit Mandelbrot

Appreciation Geometry

More Triangle Appreciation

Geometry

Today in Triangle Appreciation

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