Pick's Theorem

What is Pick's Theorem?

Pick's theorem is another method for finding the area of a polygon. This formula works as long as the polygon can be placed on a cartesian plane and the polygon's vertices has integers as coordinates. This formula was discovered in 1899 by a man named, George Alexander Pick.

Area = i + (b/2) - 1

In Pick's theorem, i represents the number of interior points  that don't connect with the polygon. b represents the boundary points. A boundary point lies on the side of the shape. So, Pick's theorem is

Area = number of interior points + (number of boundary points/2) - 1

Examples.


Green Shape:
Pick's theorem

Area = 4 + (4/2) - 1
Area = 4 + 2 -1
Area = 5 units squared

Yellow Shape:
Pick's theorem

Area = 5 + (5/2) - 1
Area = 5 + 2.5 -1
Area = 6.5 units squared

Blue Shape
Pick's Theorem

Area = 2 + (9/2) -1
Area = 2 + 4.5 -1
Area = 5.5 units squared

More Examples.




A = Q + R

Shape A

Area = 25 + (13/2) - 1
Area = 25 + 6.5 - 1
Area = 30.5 square units

Shape Q

Area = 10 + (12/2) - 1
Area = 10 + 6 - 1
Area = 15 square units

Shape R

Area = 11 + (11/2) - 1
Area = 11 + 5.5 -1
Area = 15.5 units squared

Check
A = Q + R
30.5 = 15 + 15.5
30.5 = 15.5

Fourth Demension and the hypercube

The idea of the fourth dimension and the hypercube is very hard to wrap my mind around. The idea that squares can go farther than cubes and rectangles farther than prisms. To me the images of hypercubes and tesseracts look like a smaller cube inside a larger cube connected at their vertices. To make a hypercube you have to dilate a cube. The same goes from a square to a cube. Iwonder what would happen if you dilate a hypercube. Is there any such thing as a 5th dimension?

Image of hypercube.

This next image shows what a net of a hypercube would look like. When you unfold a cube to get the net the shape goes from 3-D to 2-D. When you unfold a hypercube to get a net the same thing happens. The shape goes from 4-D to 3-D. I also think it is really cool that both the net of a cube and of a hypercube are very similar. I feel that one of the few differences is the fact that the hypercube net is just the 3-D version of the cube's net.

Image of hypercube net.