You must slice a cone parallel to the base to create two pieces such that one piece has twice the volume of the other.  If the cone has a height of h, where do you make the slice?  There are two answers.

100 Incredible Open Lectures for Math Geeks

1. Teacher Soundboard
   Add  random sound effects to your instruction.  Negligible educational value.
   
2. Brown Blobs
   Knock the brown spots off the graph with an appropriate linear equation.
   Let me know what happens when you get to level 100 – no one has gotten there yet.
   
3. Golden Spirals
   Explore the sunflower’s perfect seed arrangement.
   
4. Trig Explorer
   Examine the relationships between the three trig functions.
   
5. Slope Explorer
   Guess the slope of a random line segment.
   
6. G-Map Concept Mapper
   Quickly generate a concept web
   
7. Classroom Timer
   Just a huge timer to show on your screen.
   

A – Abel

B – Bernoulli

C – Cauchy, Cantor

D – Descartes

E – Euler, Euclid, Einstein

F – Fermat, Fourier

G -Gödel

H – Hilbert

I -

J – Jacobi

K -

L – Lagrange, Laplace

M – Mandelbrot

N – Newton

O -

P – Pythagoras

Q -

R – Ramanujan

S -

T – Turing

U -

V -

W -

X -

Y -

Z – Zeno

[A professor] likes to impress his students with the power of large numbers by drawing a line designating zero at one end of the blackboard and another marking a trillion on the far side.  Then he asks a volunteer to draw a line where a billion would fall.  Most people put it about a third of the way between zero and a trillion, he says.  Actually, it falls very near the chalk line that marks zero.

How Many Millions are in a Trillion? from Econ4U on Vimeo.

I was wearing a shirt with one of Ramanujan’s incredible formulas for 1/pi:

I was able to explain to him what a factorial is (6! = 6 * 5 * 4 * 3 * 2 * 1 = 720) but ran out of time before getting to double-factorials:

So, 6!! = 6 * 4 * 2 = 48, while 7!! = 7 * 5 * 3 * 1 = 105.  While double factorials look more impressive, they are always less than or equal to normal factorials.

I learned that double factorials are one case of multifactorials, where the number of exclamation points indicates the difference between factors:

8!!! = 8 * 5 * 2 = 80

10!!!! = 10 * 6 * 2 = 120

Looking at the entries for “factorial” on Wikipedia and Wolfram, I was surprised at how little I know about them.  I did find a few cool facts to take away, though:

• the factorial function growth faster than any polynomial or exponential function, but it can be approximated by an expression involving e and pi:
  
• the gamma function is just like a factorial but can take real and complex arguments (not just integers).  The gamma function does not return 0! = 1, so it must be defined as a special case.
  
• 0.5! = sqrt(pi) / 2

What other stuff is cool about factorials?

Found at http://www.geohive.com/earth/gen_popsize.aspx

I have recently become aware of another ratio with similar properties.  Computed as the mean of 1 and 3, or as the ratio of successive terms in the sequence {2^n}, I have dubbed this number the Wooden Ratio (after the two-by-four).  It begins 2.000… and we can represent it with the symbol Þ (thorn).

While the Golden Ratio appears in architecture such as the Parthenon, just think about how many Wooden Ratios appear in all of the brick buildings made by humanity.

In each of these bricks, the length is Þ times the width, and the width is Þ times the thickness.  Incidentally, these bricks are what we might call Wooden Rectangles.  Like the Golden Rectangle, they appear in beautiful works of art, as well as in the proportions of the beautiful face.

Notice how Da Vinci placed Mona’s Left eye at the position w/Þ, where w is the width of the canvas.

Tyra Banks, aguably America’s top model, has features which perfectly fit inside a series of Wooden Rectangles.  Notice how her mouth and eyes both show the ratio Þ.

We often see the Golden Ratio in flowers and seeds.  But look at these trees.  Enough said.

While the Golden Ratio is tied to the self-similar logarithmic spiral of a nautilus, the Wooden Ratio turns up in any natural form with bilateral symmetry.  It is also evident in the fractal formed by successive divisions of a segment into Þ segments of equal length (affectionately dubbed “abacaba“), and in the size of computer storage by repeated multiplications of the byte by Þ.

Where else might the Wooden Ratio turn up?

While looking at the graph of a parabola on Geogebra, I noticed that you could “flatten it out” by changing the x-scale on the axes or by zooming in  (changing x- and y-scales proportionally) on the vertex.  This is intuitive: if you zoom in on any smooth curve, it straightens out.

But this means that any two parabolas can be made to look identical just by zooming (as opposed to scaling only one axis), which is how two similar objects can be made to look identical.

We say two object are similar if their corresponding lengths form proportions (pairs of equal fractions); can we prove that two parabolas are similar in this way?

Say you have two parabolas, y = ax² and y = bx².  To show that two shapes are similar, we find a scale factor (k) so that any length on the first shape equals k times the corresponding length on the second shape.  For parabolas, let’s use the pair of lengths x and y for convenience.

On the first parabola, this pair of lengths is x1 and ax1².
On the second parabola, the pair of lengths is x2 and bx2².

We must find a k so that x2 = kx1 and bx2² = kax1².
Substituting for x2, we have b(kx1)² = kax1².
So bk²x1² = kax1².
Dividing, we have bk = a, or k = a/b.

So the scale factor between any two parabolas y = ax² and y = bx² is a/b, conveniently.

What would appear to be a very simple program has a surprising amount of math involved.  When a key is pressed, a letter is generated that has a random position, rotation, and speed.  These values are functions of time, and are recalculated 24 times per second.  The main portion of the program, which performs the tasks mentioned, looks like this:

createLetter = function(txt){
 if(counter++ > 100){counter = 0;}
 nm="letter"+counter;
 _root.attachMovie("letter",nm,counter+10);
 //random color
 col = getRandomColor();
 _root[nm].tf.html = true;
 _root[nm].tf.htmlText = "<font color='" + col + "'>" + txt;
 //random position
 _root[nm]._x = Math.random()*screenWidth;
 _root[nm]._y = Math.random()*screenHeight;
 //target position: center of screen
 _root[nm].xTarget = screenWidth/2;
 _root[nm].yTarget = screenHeight/2;
 //random speed
 _root[nm].speed = Math.random()*8+1;
 //random rotation
 _root[nm].embedFonts = true;
 _root[nm]._rotation = Math.random()*360;
 _root[nm].rotSpeed = Math.random()*20 - 10;
 _root[nm].onEnterFrame = function(){
 //shrink away to nothing
 this._xscale = this._yscale -= this.speed;
 this._rotation+=this.rotSpeed;
 if(this._xscale < 2){removeMovieClip(this);}
 //approach target position
 this._x -= (this._x - this.xTarget)/this.speed;
 this._y -= (this._y - this.yTarget)/this.speed;
 }
}

Quite a lot of math goes into even simple computer graphics.