Friday, August 3, 2007

The math of making things rigid.

If that's what you're into. No judgements.

A four sided geometric shape, known in math as a quadrilateral, is not by its nature a rigid shape. If such a shape is rigid in real life, that is completely dependent on the strength of the materials used. If you use 2x4's and strong nails to build a rectangular frame, it should stay rectangular. If you use toothpicks stuck into little balls of clay, your nice 90 degree angles could easily change, and your rectangle turn into a parallelogram. Also a valid geometric lifestyle choice. No judgements.

If you want to keep a quadrilateral rigid, a diagonal crossbar is the perfect addition. Now you technically have two triangles, and triangles ARE rigid by their nature, at least in the two dimensional world they inhabit. If you have a grid made up four sided objects all lined up, a diagonal in each one is the surest way to keep your grid from sliding around. (Here I drew them as squares, but they could be rectangles or parallelograms or whatever. You know what a loosey goosey secular progressive I am about this already.)

Maybe for some reason you need some of your four sided objects to be empty, for reasons of cost or weight of the total structure or aesthetics. Can the whole structure still be rigid geometrically? The answer is yes, if that's what you want, and you can test the shape before you build it.

Draw two sets of dots, one set representing the rows of the grid and the other representing the columns. If there is a diagonal strut in row 1, column 1, draw a line between the dots that represent row 1 and column 1. After you draw all the lines representing all the struts, see if you find a path between any two given dots in your diagram. If the answer is yes, the grid you construct will be rigid.

In the second 3 row by 4 column example, the resulting grid is not geometrically rigid. The third row is connected to the second column and fourth column, but not to the rest of the dots. (I drew the lines that connect those dots in red, while the other lines are black.) If you build this out of those cool Atomic Magnetz toys you can get at many cheap toy places, even supermarkets sometimes, you will find this can bend without breaking. If that's what you want. Again, no judgements.

(Blogger's note: I planned on writing this one this week before the bridge in Minneapolis fell down. The problems there, of course, deal with the necessity of rigidity for strength and forces of pressure from different angles and flexibility to deal with vibrations. Rigidity and flexibility have to be balanced. Much trickier problem to solve than this simple little puzzler that turns into an elementary graph theory problem.)

And with that, let's go to this Friday's Random 10.

You Belong To Me Vonda Shepard
Downtown Train Tom Waits
Your Feet’s Too Big Fats Waller
Forever Blue Chris Isaak
Bird Dog The Everly Brothers
Awara Hoon 3 Mustaphas 3
I Can’t Stand Up (For Falling Down) Elvis Costello & The Attractions
Let’s Get It On Marvin Gaye
A Message To You The Specials
Black Is The Color Nina Simone
Bowling Green Rafael Boguslav

Yes, that's Vonda Shepard from the Ally MacBeal soundtrack album. Yes, I know that's not what the cool kids listen to. I'm hoping to balance out some of the non-coolness with 3 Mustaphas 3 and the fact that I've got Raphael Boguslav on my iPod and all y'all don't!

But then again, no judgements.


dguzman said...

I kinda dug Vonda Sheppard too, even though I hid this fact from the cool kids.

KELSO'S NUTS said...

Great blog name also. I loved that expression. Used it myself when Chinese premier called Ohmert in for a little chat after he threatened to bomb Iran.

You have 3 of my all-time favorites on the list: Downtown Train, Let's Get It On, A Message To You. All on IPOD. Going to close my eyes now and let the songs come to me:

"I'm So Bored With The USA" THE CLASH

"Eve Of Destruction" JOHNNY THUNDERS cover

"Consolation Prizes" IGGY POP & JAMES WILLIAMSON


"Gymnopedies I & III" ERIK SATIE (comp)

"She's A Rebel" GREEN DAY

"Little Boxes" PETE SEEGER cover

"Angel Of The Morning" JUICE NEWTON


"Slapshot Man" THE ZAMBONIS

Matty Boy said...

Thanks, KN. Nice list o' tunes as well. I stole^H^H^H^H^H borrowed the idea of Random 10 from my bandmate Padre Mickey, and I'm sure it's not original with him. If you want to Random 10 on your own blog, be my guest.

FranIAm said...

Trivia fact- Padre Mickey and Kelso are both living in Panama.

Bonus maybe fact- didn't you once live there too Matty?

Matty Boy said...

No, I didn't live in Panama. I taught in El Salvador.

Nice to have another reader from Panama. I thought the adress looked different.