Friday, March 06, 2009

Tesselation and angles

These are a couple of Sean's messy drawings of tessellating shapes.

He told me the other night that the only regular polygons that tessellate are the triangle, the square and the hexagon. It had something to do with the interior angles of these polygons. I didn't understand what he meant and called for Brian of course.

Brian explains that only regular polygons with interior angles that are factors of 360 tessellate.

The boys watched me as I tried to find out the interior angle of each polygon, to see if it was really true that only the interior angles of the triangle, square and hexagon are factors of 360.

The interior angle of an equilateral triangle is of course 60deg, so that goes into 360deg 6 times. Tick.

The interior angle of a square is 90deg, and that goes into 360deg 4 times. Tick.

No protractor in sight, I ask Brian, er, what's the interior angle of a hexagon?

He tells me this:

So to find the angle of any regular polygon, you've gotta subtract the number of sides by 2, multiply by 180deg, and divide by the number of sides.

After my usual "are you sure?", I got hold of a calculator and started doing what Brian said. First, I checked if the formula worked for a square, it does.

So I find that the interior angle of a pentagon is 108deg, and that doesn't go into 360deg. Which means it doesn't tessellate.
Moving on to hexagon, the angle is 120deg, which goes 3 times into 360deg.  Tick.

We've got all 3 of the regular polygons that Sean tells me are the only ones that tessellate.  But do I stop?  Nooooo....I'm anal that way....I continue calculating for the rest of the whatever-gons.  

I was about to calculate the interior angle of a 14-sided regular polygon when Brian asks me, "What are you doing Mummy?"

Me: The next number that will go into 360deg is 180deg, so I'm looking for the polygon with that interior angle.

Brian: You can't!  180deg is a straight line!

Me: Uh...oh...hmmm...really?  

I then calculated the angle for a 1,000,000-sided polygon, and the calculator showed, "179.99964".

Ohhhhh, you're right Brian, I said, feeling really dense. Unlike me, Brian doesn't gloat.

I can't believe I was gonna go on calculating interior angles of goodness knows how many more polygons. 180deg! Of course it's a straight line, what a Duh! mummy moment. Unbelievable.

*Note: Sean's first drawing shows tessellation of an octagon, but it only works if combined with a square of equal sides to the octagon's sides.


Alcovelet said...

You're too funny, 180 degrees! And what a good natured guy Brian is about not gloating :D. So fascinating, this. Thanks for this bite-size and fun bit of education!!

Anonymous said...

Can see that Sean is going through the MM craze. I can relate to that! LOL!!!

monlim said...

I think we only do tessallations in p3. Sean is really advanced! And bravo to you for trying out everything! I would've just taken Brian's word for it.

Lilian said...

Adeline: Brian is very nice to everyone except his own little brother. If it were him doing the calculation mindlessly, I would definitely have scolded him, "Aiyohhh!!! Brian!!! Why never use your brain!!! Don't you know 180deg is straight line???!!! Seelah, that's what happens when you play too much xbox!"

Sandy: Yeah, this is Round 2...Brian had the craze some years back and now it's Sean's turn. The books have paid for themselves MANY times over.

Monica: Nolah, when Brian was younger, we found tessellation picture books in the preschool section of Singapore's library. I tried to calculate cos thought maybe I will have a "Eureka" moment...end up having a "DUH!" moment.

bACk in GERMANY said...

Haha... 180 deg!!!!! Of course it's a straight line. No wonder you can't sleep at night.

But har... what tessallations? Never heard of the term! And this is, err what, in the P3 syllabus???!!!!

Bring the boys to Barcelona's Science Museum during the spring break! Introduce them to Gaudi's hexagonally paved sidewalk.

Oh... btw, Brian's so sensitive to mommy's duh moment... Brilliant IQ with an even more brilliant EQ. No wonder he draws friends to him wherever he goes!

Lilian said...

PP: You know what tessellation is lah, even if you've never heard of the term. Think of honeycombs, that's tessellation lor. Geddit?

You keep enticing me with case you haven't noticed, there's a worldwide recession now. Oh, I forgot, you and your husband both work for gahmen, recession-proof! (don't hantam me, I'm just saying what everyone's thinking :) )