Awesome Quote

Just had to post this, it's so awesome. A quote from one of my friends who I'm in a play with, this is how he checked in.
"I was going to procrastinate today, but put it off until tomorrow." -H.H.

Simultaneous Equations

I love Simultaneous Equations. It's so cool how you can take two completely random equations (2x + 8y = 20 and 2x + 4y = 12 for example) and just add or subtract them (2x + 8y = 16 - 2x + 4y = 12 comes out as 4y = 4, therefore y = 1 and plugging that in, 2x + 8 = 16, 8 + 8 = 16, 8/2 = 4, therefore, x = 4, so the answer is: (4, 1)) before you know it you have a pair of coordinates. Here's something really cool though. Say you have a pair of simultaneous equations like this: x + y = 15, y = x +3. That is telling you that y is the same number as x + 3. So, going back to x + y = 15, plug this in, it will be, x + x + 3 = 15, simplifying, 2x + 3 = 15, subtract 3, 2x = 12, divide, x = 6. Plug that in 6 + y = 15, 6 + 9 = 15. Therefore, the answer is: (6, 9). Here's a weird thing about simultaneous equations: say you have two equations, 5x + 3y = 35 and 5x + 3y = 33. Solving by subtraction you get 0 = 2. That can't be right. This type of simultaneous equation is called an inconsistent simultaneous equation. Another kind is: say you have x + y = 11 and 3x + 3y = 33, solving, we can multiply the first equation by 3 (another operation you can perform on equations) and then you have 3x + 3y = 33 and 3x + 3y = 33. Solving by subtraction we get, 0 = 0. Well thats not very helpful. These types of simultaneous equations are called equivalent equations (reheheheheheheheheheally?). That's all I have to say about simultaneous equations for right now. But here's one to work on. (See top right corner ;))

Graphing With Curves

So, everyone knows that if you draw an x-y axis and put two points on it, you can make a line, therefore, making a line graph. But how surprised was I yesterday, when I was doing my math as normal, solving simultaneous equations and determining whether they were inconsistent or equivalent, when I come upon a most curious thing, a curve graph.




Now, I had never encountered one of these before, but it turns out that, say you have an equation: 3x + 7y = 21. To graph this as a line you would just find the the x and y intercepts (7,3) and then draw a line between those points and extend it. But, turns out, if you square x and y, the graph will become a curve. So, change it to: 3x^2 + 7y^2 = 21. The graph of that is a curve.

Physics, and Geometry

Physics of Pruney Fingers Revealed

• By Lisa Grossman  
• March 8, 2011  |  
• 7:19 pm  |  
• Categories: Physics

An exploration of mathematical shapes could explain why skin gets wrinkled after too much time in the tub. Understanding the geometry of wrinkly skin could help design new materials that can stretch out without losing strength.

“The paper explains a mechanism that can explain the structural stability of keratin in skin and its ability to absorb very large quantities of water,” said mathematician Gerd Schröder-Turk of the University of Erlangen-Nürnberg in Germany, who was not involved in the new work. “This is a major breakthrough.”

Scientists and frequent bathers know that skin can absorb a tremendous amount of water, and still be a strong barrier between our bodies and the harsh outside world.

“Your skin wrinkles, yet it maintains its structure,” said mathematician Myfanwy Evans of the Australian National University, lead author of the new study. “It doesn’t just fall apart and dissolve into the water.”

The skin’s resilient stretchiness comes from an intricate network of fibrous proteins called keratin, which make up the outermost layer of the skin, as well as hair and nails. Scientists knew that skin’s keratin networks were important, but the arrangement of fibers was uncertain.

Now, Evans and Australian National University colleague Stephen Hyde may have found a solution. They describe their stringy skin model in the March 8 Journal of the Royal Society Interface.

“It explains a lot of mechanical features that hadn’t really been able to be explained before,” Evans said.

The researchers stumbled upon the new model in a purely math-based search for interesting topological shapes. Evans studies a class of beautiful mathematical shapes called Gyroids, which show up all over the natural world, from lipid membranes to butterfly wings.

“It’s an interesting fusion of maths and experimental science,” Evans said. “These are popping up everywhere.”

Using computer simulations, Evans and Hyde explored what would happen if you took infinitely long threads and wove them through the labyrinth of the Gyroid surface, then took the surface away. Some of the resulting 3-D woven structures were so tangled that none of the threads could move without breaking the connections between individual threads. If keratin were arranged this way, Evans says, our skin would lose its strength when it got wet.

“Losing contacts between keratin fibers means losing structural rigidity,” she said.

But other weavings could expand, with threads straightening and sliding along each other without losing contact. One of these, which Evans and Hyde call G₁₂₉, could swell to fill a volume seven times greater than its original shape, while keeping all its fiber connections intact — just like skin.

Suggestively, the model’s version of keratin networks in dry skin matches real data almost exactly, Evans says.

“That was quite convincing evidence that it’s highly likely that this model really does work,” she said.

Although the model hasn’t made it far from the world of abstract math, Evans and colleagues hope their models of expandable networks of fibers could be used in the bottom-up design of custom materials with controllable stretchiness. These materials could be useful for things like bandages, bulletproof vests and artificial skin, she suggests.

“This could be a really good target for bio-inspired materials,” she said. “It’s not a matter of testing it in the lab, it’s a matter of understanding its geometry in order to understand its physical properties…. We hope this paper will put that idea out there, and maybe lead to some new interesting materials.”

Cool Link!

Awesome demonstration of the elements. Fun to watch too.
http://www.periodicvideos.com/#

One Thick Textbook

Sorry, I had the flu. Here's a picture of an ultra-thick textbook.

Hello World!... again

Copy and paste this into Python, or Pygame if you have it.



import pygame, sys
from pygame.locals import *

# set up pygame
pygame.init()

# set up the window
windowSurface = pygame.display.set_mode((500, 400), 0, 32)
pygame.display.set_caption('Hello world!')

# set up the colors
BLACK = (0, 0, 0)
WHITE = (255, 255, 255)
RED = (255, 0, 0)
GREEN = (0, 255, 0)
BLUE = (0, 0, 255)

# set up fonts
basicFont = pygame.font.SysFont(None, 48)

# set up the text
text = basicFont.render('Hello world!', True, WHITE, BLUE)
textRect = text.get_rect()
textRect.centerx = windowSurface.get_rect().centerx
textRect.centery = windowSurface.get_rect().centery

# draw the white background onto the surface
windowSurface.fill(WHITE)

# draw a green polygon onto the surface
pygame.draw.polygon(windowSurface, GREEN, ((146, 0), (291, 106), (236, 277), (56, 277), (0, 106)))

# draw some blue lines onto the surface
pygame.draw.line(windowSurface, BLUE, (60, 60), (120, 60), 4)
pygame.draw.line(windowSurface, BLUE, (120, 60), (60, 120))
pygame.draw.line(windowSurface, BLUE, (60, 120), (120, 120), 4)

# draw a blue circle onto the surface
pygame.draw.circle(windowSurface, BLUE, (300, 50), 20, 0)

#draw a red ellipse onto the surface
pygame.draw.ellipse(windowSurface, RED, (300, 250, 40, 80,), 1)

# draw the text's background rectangle onto the surface
pygame.draw.rect(windowSurface, RED, (textRect.left - 20, textRect.top - 20, textRect.width + 40, textRect.height + 40))

# get a pixel array of the surface
pixArray = pygame.PixelArray(windowSurface)
pixArray[480][380] = BLACK
del pixArray

# draw the text onto the surface
windowSurface. blit(text, textRect)

# draw the window onto the screen
pygame.display.update()

# run the game loop
while True:
    for event in pygame.event.get():
        if event.type == QUIT:
            pygame.quit()
            sys.exit()