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Thursday, December 30, 2010

I was a paperboy since before I could walk

I have LOVED origami since I was a wee lad. That said, it got a little boring for me. Not so boring that I wanted to quit, just boring enough that I haven't done much for a while. However, I recently experienced a spike in my origami excitement over the last year or so:
Most people are at a 3, true experts range from 50 to 75
I was either not yet born or too young (or both) to appreciate origami in the early 80's After that, things sort of progressed in a nearly strait line until 2005.

This is because of a few things: firstly, I wrote a paper on the subject for my History of Math class in the spring. Secondly, I decided to apply for a research opportunity at the university focusing of origami. I have a few questions to get started if they like my proposal:


The area of a square piece of paper is fixed and cannot change (since cutting and pasting/taping is illegal) but as you fold, the part you see will change. Of course, you can never make more area than what you started with. The perimeter also changes but does it have an upper limit?
So along the way, someone figured out that there is a way to make a shape have an infinite perimeter but never exceed a certain area (remember perimeter is a measurement of the edges of a shape, area is a measurement of the inside of a shape)
This was a really quick sketch I made to illustrate my point: the circle is this fixed thing. The area and perimeter do not change. You can measure them, write home about them but that would be boring to read (and to write) but the triangles are VERY interesting. This was just three levels I drew but I think it is enough to see something interesting: you draw that first big triangle and make it touch the circle at all tree corners. Then, for every side of a triangle, draw another so that one side (of the new triangles) touch the first big one. Now there are 12 "sides" where you can draw even more triangles on those you have already got! Repeat until you die* and you will have accomplished something very neat: That weird jagged shape you made will always fit in that circle so the area MUST be smaller but the perimeter is MUCH BIGGER than the circle's! There is actually no upper limit to the perimeter so you could make it infinite if you lived that long.

*I really hope you have better plans than this with your life though. Sheesh! What is wrong with you anyway? You have so much more potential!

Is origami the same as this drawing? Can you make a flat shape by folding, which has a larger perimeter than that square you started with? (And fyi: always a square in traditional origami) I personally don't know the answer yet (hence a good area to start my research right?)


Is there a reasonable way to construct the solution to the general quintic equation? This one is probably beyond me since it has been beyond EVERY SINGLE MATHEMATICIAN EVER (and everyone else)

What is a quintic equation you ask? Well lets just say it is a squiggly line represented by an equation that looks something like this:
 You would read this out loud as "y equals x to the fifth" (or rather, I would. and have. and still do.) This is probably the simplest quintic equation and doesn't scare too many people but they can get a lot harrier. And gross. You could also add an "x to the fourth" and "x cubed" etc. You could also multiply any of those individual terms by a regular ol' number (or imaginary number. imaginary numbers are actually just as real as regular numbers . . . which isn't saying much since numbers were just imagined in the first place. So basically, numbers were never real to begin with)
Note: actually my explanation is wrong since the squiggly line is the representation. The y=(x-stuff) thingy is the actual thing. (well sort of.) But hey, this is my blog and I'll misexplain if I want to. (♫ you would misexplain too if it happend to you ♫)
 There has been some work in mathematical origami that has proven that if you make so many simultaneous folds, there exists a way to solve the general quintic equation . . . but nobody has quite figured what that might be. I'd like to work on that too.

And just for fun, I want to tell you about some neat things that the ancient Greeks would be jealous of: First, you can easily use origami to divide ANY angle into thirds. Second, you can easily make two cubes where one is exactly twice the volume of the other. (well sort of, you can find the lengths of the sides anyway) Man. Those ancient Greek mathematicians and their crazy antics! Paper was invented 2000 years too late in my opinion. Why were these guys drawing stuff with their straightedge and compass when they should have been inventing paper? Certain wasps had got it figured out by then I am sure. But Pythagoras and Euclid were probably too busy living it up to get anything else done.

Perhaps I will save those origami tricks for another day. They really are quite neat and fun to do! If you want, you can find them in this book.


Sunday, December 19, 2010

Induction: for really tall buildings

Prologue: In which there is a picture of a ladder, a metaphor is made, and Dashbo asks you to do a lot of senseless adding.

This is induction
Induction is a ladder. This ladder has some interesting properties to it. First, you can always get to the bottom rung on the ladder. Second, if you can get to any rung on the ladder, then you can get to the next rung as well. So does it make sense that you can get to any rung on the whole ladder? (because you can get to rung 1 and that means you can get to rung 2 and since you are there, you can get to number 3 etc.)

Induction is a ladder that proves some kind of math-something-or-other. And making this ladder takes two steps. (steps is a word which here means "any maneuver made as part of progress toward a goal") but once you have established this ladder, it usually has an infinite number of steps. (steps is a word which here means "rungs on a ladder")
Have you ever had to add all the numbers from 1 to one gajillion? Well then induction is the ladder for you! But the really slick part is that you don't need to actually climb the infinite latter, just make it and walk away!
Let us start by adding something smaller from 1 to . . . 1. Yes all the numbers from one to one should be fairly easy to add. it is just 1
lets do another slightly complicated one: 1 through 2
1+2=3     Done and done!
lets keep going a little bit more
Is there an easy to see pattern? Well, there is and some guy named Gauss figured it out when he was a baby or something. He wasn't the first, but later in life, he drew a really famous 17-sided polygon so we give him the credit. Today, I could show you how to draw a picture that makes it really obvious but I am not going to. Instead, you get the inductive proof. If you want to see Gauss's way of thinking about it, maybe you should start your own math blog. I won't be offended.


To see the pattern, lets call the last number that we add "n." So we want to add from 1 to n and it just so happens to be
Are you going to take my word for it? Well you whatever your answer is to that question is, I am going to show you the proof so it doesn't matter what you think.
Now on to the two steps of building an inductive proof!

Chapter 1: The Base Case This is a short chapter in which we get to the first rung on the ladder and add all the numbers from 1 to 1. Yes, again.

In the base case we have to get to the bottom rung. So let us just try it with our little formula for n=1. We already know that it equals 1 because we added it in the prologue. but just for fun, we'll do it the other way:
Oh good! It works for the first rung in our ladder! And since numbers don't really ever change, this is always true for ever and ever so far as I can tell.

Chapter 2: The Inductive Step In which we climb the ladder from any rung to the next rung up and somehow climb the whole thing without actually climbing at all.

 Now is the tricky part. We need to show that going from one step to the next is always going to be the same thing. We start with an assumption:
Suppose our formula is true for any number n. I already told you it is true so you might have already assumed it is true. Remember this image?
It is a rerun from earlier in the blog post
I do. I got it from Wikipedia
well if that is true then climbing to the next rung is to try "n+1"
using our keen algebra skills, lets add n+1 to both sides of the equation. 
We are almost done! If all the junk on the right-hand side can be put into a form that looks like our formula, then it is done and we can go home! (with new knowledge!!!)
to do that, lets take our part we just added and multiply by 2 divided by 2, (2/2)
This lets us turn the whole mess into one fraction since they now have a 2 on the bottom.
Notice how there is an "n" and a "2" on top that are both multiplying an (n+1)? We want to get those (n+1)'s together. We do this by factoring. Remember the water-bottle example? If the 2 and the n are both being multiplied by an (n+1) then the rules say it is 100% legal to stick them in the same package and just multiply that by the (n+1). This is the result:
 So now, it is super easy to take that 2 on top and turn it into a 1+1. It will make that part of the equation look like (n+1+1) And finally, we just add some more plastic wrap to that and exclude one of the 1's (since one is the loneliest number anyway.)
Whoa! Did you see the magic happen? We said in that rerun image that 1 to a number n is that neat little formula. We didn't prove THAT part, we just assumed it to be true. But from that, we showed that we can add the next number (n+1) and what did we end up with? THE EXACT SAME FORMULA BUT FOR THE NEXT NUMBER!!!!!

Epilogue: In which we reflect on the significance of something.

In the Inductive step we really only proved that IF true for some number, THEN it is true for the next number. This really doesn't mean much unless you remember the base case.
But it IS true for n=1 and so it MUST ALSO BE TRUE FOR 2 AND 3 AND 4 AND . . . EVERY WHOLE NUMBER BIGGER THAN 1!!!!!

Let us reflect on the significance of this: We can now solve our "add all numbers from 1 to one gajillion" problem. it would be (one gajillion) x (one gajillion + 1) divided by 2. HOORAY FOR ENDLESS MATH LADDERS!!!

That wasn't too shabby was it?

Sunday, October 17, 2010

Playing GOD (it is what mathemeticians do!)

There is an interesting branch of mathematics called set theory. When math people say "theory" it actually means a whole branch of mathematics that has its own rules that math people get to make up themselves. The rules are called "axioms."

Can life possibly get better than being aboard the Axiom?

In fact, you can make your own math if you want. Just imagine a new world or universe and make up your own rules.

Dashbo's world: (or we could say Dashbo theory)
1. Gravity works the same as on Earth except that it pulls things to the right instead of down.
2. The only primary colors are yellow, pink, and green.
3. When things melt, they change color by losing the yellow they might have had. and when they evaporate, they change color by losing whatever green hues they might have had

You get the idea. Once I pick all my rules for my imaginary place, then I take an imaginary object and put it in there and see what happens:

Lets say my object for sure has a right and a left side. Well then we know which direction gravity will be pulling it. If I spin my object, then gravity will do all sorts of weird things to it. I imagine it will probably make a spiral path outward. We call that a conjecture. But can I prove it? If I can, then it is no longer a conjecture, but now it is a theorem. Have you ever heard of the Pythagorean theorem? That is something that has been proved always true (in the imaginary world that some math dude invented called euclidean geometry) Euclid had five rules that he made up. Math people spend a lot of time trying to prove conjectures into theorems or trying to take theorems and re-prove them in new ways. And the only thing you get are the axioms at first. (but if somebody proved a theorem from the axioms, then you can always use that theorem to prove your theorem.)

So can you prove an axiom? NO! axioms are unprovably TRUE for your world! They just EXIST. Trying to prove them is like trying to tell a deaf person what the note B flat sounds like. If you said to yourself: "well he might not have always been deaf, what about Beethoven? He was a deaf composer dude!" Then you should probably go to someone else's blog. If this was a building, then the axioms would be the most base fundamental bedrock that you build everything else (theorems and conjectures) on. If you take your world and start chucking axioms, then your theorems fall down in a heaping crumbly (and probably toxic) mess.

Set theory is a math world based off of real life situations. You have heard of a set right? (I heard once that in English, "set" has the most definitions than any other word. You can set the table, you can set something up, get ready, get set, go!) I am talking about the kind of set that is a collection. Like your mom's set of fine china or your set of matching Elvis figurines. So set theory's axioms or rules are based off of what people noticed about sets. Can one object be in more than one set? Can one set exist entirely in another? Can two sets be entirely separate where they don't share anything? I am pretty sure you have seen a Venn diagram. They look like this:

This is a really slick way to represent sets! You can easily tell where to look if I asked for everything in both C and B or everything NOT in B etc.

So in math, we use numbers a lot. someone decided it would be easier to make some sets out of them. here are a few of the most used number sets:
All positive numbers
All negative numbers
All whole numbers (includes positive, negative and zero)
All natural numbers (or counting numbers. it includes one and then all the whole numbers bigger than one)
Imaginary numbers (it turns out they are just as real as any other number really. Who knew?)
Real numbers (this set is really really big. In fact, some sets are infinite and yet this set is bigger than those sets)
Rational numbers (that would be fractions)

You can make your own number sets if you want. I don't really know why you would though. If you do, let me know what it is and why you decided to make that set in the first place.

Monday, September 27, 2010

Water is so abundant, but if hoarding it is your hobby, you need to know the order of operations!

I just looked up "order of operations" on wikipedia. There was some pretty good stuff there. http://en.wikipedia.org/wiki/Order_of_operations

This is an important thing to understand in algebra because if you mess them up, you get the problem wrong and someone could get hurt! If you are recklessly doing math, you become a danger to yourself and those around you. So with that said, you should probably be wondering what an operation is and why they need to stay in order - unless you already know.

( ) ^ * / + - (these are the computer symbols for the operations in the right order.)
"Operation" is a posh way to say: "parentheses, exponents, multiply and divide, add and subtract, etc." (also in the right order) These are the ones used most. (actually parentheses are not an operation, its just a way to keep organized but you can think of it as one for now)

Why do they need to stay in order?
. . .

Thats the harder part

. . .

Did you know that multiplication can be done in rows and columns?

Here is 12 cars. But it is arranged in 4 rows and 3 columns. 3x4=12. It works every time.
That is part of the topic. I promise . . .

lets say you have to figure out 3+4x8. How do you do it? do you start left to right like reading?
(the answer is no. I wanted to tell you that before I did it so that it didn't get stuck in your head the wrong way)
3+4=7 and then 7x8 is 56. So is that what you did on your last test? Well that's bad. You should've gotten that one wrong.

The order of operations says we have to do the multiplication first. Would that really change anything though??? Well lets go back to our rows and columns:
7x8= (7 rows and 8 columns)

its 7 times 8. I promise.
well what the heck! 3+4 is in fact 7 and 7x8 is 56 because just look at the stars! there are 56 of them!

Well here is the right way to do 3+4x8:

4x8 is 32.

So now we add the 3.
3+32 is 35!

So what it comes down to is you will end up with extra rows or columns and stuff. This is the way math language is written. I don't know why. I suppose its easier for math people.What about parentheses? what is up with them?

Suppose you have a strange hobby of collecting bottled water

From the looks of it, you have quite a collection! You should be proud!

Except that its so messy! Maybe you want to get them organized?

I know! Use plastic wrap!!! That stuff is great! you can use it all over the place, and they make it now so that its easy to cut and you don't have to struggle to get the piece you want :)

That is much better! I bet you can make tons of these neat little packages with all the water you have! You don't have to keep the different kinds separate you know. You could put a Dasani in with 3 Aquafina's and 6 Fiji's. or something creative like that.Its a good idea and math people do the same thing with their math. The plastic wrap doesn't weigh anything (so you can add as much as you want to those balance scales of the last blog post and it won't change anything) It looks like this: ( )

Then put the stuff inside:

(6x8+56-7+ . . . )
so if you had a math person describe your water, he'd take all the plastic wrapped stuff and write it like this:
Dasani + 3 Aquafina + 6 Fiji ) or even shorter: (D+3A+6F)now lets say you like that particular set of water a lot because it reminds you of a Led Zeplin concert you went to when you were younger. You happen to have enough water to do some more sets exactly like that with a nice fresh roll of plastic wrap. In fact, you can do 3 more just like it and so now you have 4 total!
(D+3A+6F) + (D+3A+6F) + (D+3A+6F) + (D+3A+6F)That took some serious copy and paste action on my part and if I was doing it by hand, it would take too long because I am hungery right now. Its easier to say 4(D+3A+6F). Its a multiplication problem where we have 4 rows and one column of water packets (and we made the "x" sign invisible probably because it means death if drawn on a cartoon character's eyes and that would've been too confusing while doing math on Saturday mornings)

but if you were to take off all the plastic wrap (even though it was a lot of work) what would you have?

It looks like
4 Dasani's, 12 Aquafina's and 24 Fiji's. Its the same water, nothing changed in your collection, its just organized differently. In the order of operations, you do parentheses first because if not, your water would be counted wrong:
4(D+3A+6F) Remember that there is an invisible "times" sign so if you aren't watching Saturday morning cartoons, feel free to actually write it out.
4x(D+3A+6F) so we already discussed how much water we've got. And if you forget the order of operations, you would think ". . . so I multiply 4 times my Dasani's and then I just add my three Aquafina's and six fiji's and I'm done. That is how much water I have. . ." and you'd be wrong. that "4" means four Led Zeplin concert packs; not just four Dasani's!
What if you had a Led Zeplin concert pack and some other package that you give away at Christmas? could you wrap them separately and then wrap them together?

So then what is this equal to?

Monday, September 13, 2010

Attack the killer-psycho-maniacs that teach mathematics!

What does the word "algebra" make you think of?

Do you picture your math teacher? What does he look like?

Is he just as evil as any horror movie psycho-killer and yet still makes Ben Stein seem like America's next runner up on Last Comic Standing? Have you ever caught him or her plotting another twisted homework assignment that will take hours but not teach you anything?

"Welcome to the government-style-K-through-12-approved-mathematics-curriculum-for-brain-washed-vegetables . . ." he says on the first day of class. Not with words maybe. But you know he was thinking it (and grinning on the inside) . If you know exactly what I am talking about, keep reading. This blog is for you! Those guys have turned math mind-numbing piece of muck. Or at least they try as hard as possible to make it look that way.

My opinion on algebra now is that I was never taught why it is so amazing. It was too hard to "see" since it was just letters and numbers with one of these symbols: =, >, < etc. Most of the K-12 math teachers and teaching that I experienced was very left-brained and technical. Right-brained folk like me hardly stand a chance.
My hope is that I can take the technical left-brained stuff that I was taught and turn it into pictures, stories, and anything else from the artist's world. I am currently attending college working toward a BA degree in mathematics. Why? because after algebra, I took trigonometry and calculus (right-brained stuff like that geometry class you liked) from an amazing teacher who helped me visualize it. I discovered how creative and fun this stuff was. I just had to muddle through the first 18 years of the pre-fun math.
So . . . with all that stuff out of the way, I want to tell you how I picture algebra now.

Algebra looks like this:

(except I don't picture the hanging chains holding the baskets, just bowls bolted right to the balancy-stick)

In class, picture it. But instead of drawing out the whole thing, your math teacher wants you to use a less-fun symbol:
  • if it balances, you say that with one of these: =
  • if the left side is heavier, go with a "greater than" >
  • if right is heavier, then use "less than" <
Did you read those bullet points with your best Ferris Bueller's Day Off voice? Because you can if you want. I won't be offended. But just so you know, That is almost the whole of what algebra is. Once you got that down (and it took me a long time to realize this is what it is) you'll know probably most of what is going on in class.
  • Just so you know, if you take something out of one bowl, take it out of both bowls. ( ". . . subtract [whatever blah blah blah] from both sides . . .")
  • Same with putting stuff in the bowls. This is a really strong scale. It can hold heaps and heaps of stuff as long as you put it in both sides.
  • If its already not balanced, you still might have to add the same amount to both bowls. When it is tipped to one side, it can be something really really small and it can still looks just as tipped (tipsy?) as if there was an extra elephant in that bowl.
  • Most of the time, you don't want to change the position of the scale. (don't go from balanced to unbalanced or the other way around)
What do you think? Does that help?