64=65 Illusion

Take a look at the following image.

It baffled me for a good 10 minutes, and frustrated me for the next 20. It apparently shows that, using simple rules of geometry, 64=65. It is obvious that this is some sort of optical illusion rather than being an image that will shatter the foundations of mathematics, but how is it done?

The first shape is an 8x8 square, that has been split into 4 quadrants:
   Pink has area of 0.5x3x8 = 12
   Red has area of 0.5x3x8 = 12
   Blue has area (3x5)+(0.5x2x5) = 15+5 = 20
   Green has area (3x5)+(0.5x2x5) = 15+5 = 20
And so the first shape has area 12+12+20+20 = 8x8 = 64. Everything seems legitimate so far...

The second shape is a 5x13 rectangle, made using the same 4 quadrants:
   Pink has area of 0.5x3x8 = 12
   Red has area of 0.5x3x8 = 12
   Blue has area (3x5)+(0.5x2x5) = 15+5 = 20
   Green has area (3x5)+(0.5x2x5) = 15+5 = 20
But now the shape has area 5x13 = 65, but is entirely made up of shapes with areas 12+12+20+20 = 64. Thus we are forced to conclude that 64=65.

But this is clearly a load bull***t! For the next 25 minutes I thought, pondered, wondered, and started pulling my hair out at how this feat was done. So I cut out my own 8x8 square, cut it in exactly the same way as shown above, and tried to reassemble the 5x13 triangle, and found the pieces didn't fit! There was a very thin, very stretched diamond shaped gap in the middle. It was time for some calculations:

 

According to the diagram above shows that the angles a + b = square angle, and angle of 90 degrees. We have:

   a = arctan(3/8) = 20.556...

   b = arctan(5/2) = 68.199...

Therefore a + b = 20.556...+68.199... = 88.755..., and not 90. There is a missing angle of 1.245... degrees. So we must now conclude that the line that splint the 5x13 rectangle in half is not a straight line, and so the geometry falls apart. Thus, mathematics has been saved!

Gabriel's Horn

We all love a paradox, and here's one that really makes my mind boggle. Imagine a three dimensional shape with finite volume, but with infinite area, what would it look like? Are you imagining complicated surfaces that turn in on themselves in all weird and bizarre manners? Take a look at this:

This structure is called Gabriel's Horn, after the horn the Archangel Gabriel blows to announce judgement day. It is also called Torricelli's Trumpet, as its properties were first studied by Italian mathematician and physicist Evangelista Torricelli (inventor of the barometer) in the 17th century.
It is formed by taking the curve of y=1/x in the domain of x greater than or equal to 1, and rotating it around 360 degrees. Although first discovered before the invention of calculus, today, by using integration, we can show that it has a volume of PI, and an infinite surface area.

Without looking too deep into the mathematics here, the paradox can be considered from a painter's point of view. Imagine the Gabriel's Horn was a real object, we could fill it with a finite amount of paint, but we could never really fill it, as not one drop of paint would ever reach the bottom. Also, you could never paint the entire surface of the horn, although we have already filled it with a finite amount of paint. Strange.

Picturesque Parametrics

Yes, parametric equations can be beautiful! Well, the graphs produced by them can be anyway. Take the Lissajous Equations for example, a pair of simple trigonometric parametric equations, that produce an infinite amount of stunning, alluring and varied graphs, simply by changing two or three variables within the formulae.

Take a look at some of the graphs below, surely worthy candidates to be called "art"? 

 So how were these graphs created? Simple parametric equations!

x=sin(At+B)cos(Ct)

y=sin(At+B)sin(Ct)

by changing the variables A, B and C, we can produce the most magnificent and elegant of shapes and patterns. The graphs above were made using A=96 B=3 C=1, A=100 B=7 C=427, A=36 B=1 C=173 and A=46 B=22 C=173.

You can try some of these out for yourself using Excel, which is how I made these graphs.

These ones used A=7 B=5 C=80, A=7 B=1 C=93, A=76 B=11 C=57 and A=55 B=3 C=189. And here are some more:

and these used A=51 B=1 C=50, A=36 B=4 C=209, A=87 B=11 C=13 and A=58 B=1 C=475.

I hope you're as fascinated as I am with these graphs, and that it will open you up to the naturally gorgeous world of maths.