Pick's Theorum

             Recently, I was shown an amazing way to estimate the areas of shapes using something called Pick's Theorem. This theorem, created by George Alexander Pick in 1899, uses a plane with points each one unit apart, basically a Cartesian Plane in a different format, and doesn't require axes. This Theorem uses the number of points seen on the interior of a shape, and points seen on the edges of a shape. "i" represents interior, or lattice, points, and "b" represents edge, or boundary, points. The formula goes as follows: Area = i + (b/2) -1. There is also a version which incorporates interior spaces in the shape, which you can read about on this website. This theorem is especially useful for finding the areas of complicated shapes whose areas would be difficult to solve via rectification or by splitting the shape into several different shapes. Picks Theorem is also very similar to another method of estimating area, which uses a Cartesian Plane format with a grid rather than just points, in which you count the number of squares within a shape, and then count the number of squares divided or cut by the shape. Using Picks variables, "i" would be the interior squares, and "b" would be the boundary squares. By cutting the number of cut squares in half and adding that number to whole, interior squares, you can estimate the area of an object. The formula would be A = i + b/2. Below are some examples of shapes whose areas were estimated using both methods.

Picks Theorem: Area = Lattice Points + (Boundary points/2)-1

Other Method: Area = Full Squares + Partial Squares/Two

Yellow Shape (Pick's)

A = 5 + (5/2) - 1

A = 15/2 - 1

A = 13/2

A = 6.5

Other method

A = 2 + (10/2)

A = 2 + 5

A = 7

Light blue shape

A = 2 + (9/2) -1

A = 13/2 -1

A = 11/2

A = 5.5

Other method

A = 0 + (11/2)

A = 5.5

Red Square

A = 0 + (4/2) -1

A = 2 -1

A = 1

Other method

A = 1 + (0/2)

A = 1

Hexagon

A = 4 + (6/2) -1

A = 4 + 3 -1

A = 6

Other method

A = 3 + 6/2

A = 3 + 3

A =6

Gray Triangle:

A =1+(3/2)-1

A = 3/2

A = 1.5

Other method

A = 0 + 4/2

A = 2

Second green triangle:

A = 0+(5/2)-1

A = 3/2

A = 1.5

Other Method

A = 0 +3/2

A = 1.5

Green shape(R):11+(11/2)-1=15 1/2=15.5

Figure A

A =  25 + (13/2) -1

A = 25 +6.5 -1

A = 31.5

Other Method

A = 22+½(14)

A = 22 + 7

A = 31

Figure Q

A = 10 + (12/2) - 1

A = 10 + 6 -1

A = 15

Other method

A = 7+½(14)

A = 7 + 7

A = 14

Figure R

A = 9 + 12/2

A = 9 + 6

A = 15

Other Method
A = 9 + ½13
A = 9 + 6.5
A = 15.5

The Fourth Dimension

The fourth dimension is a very strange place for us to perceive. I believe it can be described quite well by this video Forget about your familiar world. But despite the simple explanations and mathematical reasons, I believe its pretty much impossible for us to image the fourth dimension. The process for imagining the fourth dimension is simple. You have a point in space. It has no dimension. You translate the point, and connect the two points with a line. That is the first dimension. You can then translate the line to get a plane or the second dimension, then translate again to get a three-dimensional shape, say a square. Translating the square gets you to the fourth dimension. We still only see this image in a three dimensional way though. As the video explains, we could only see the fourth dimension as an ever changing, three dimensional cross-section of the whole. 

Imagine you are a two dimensional being. Also difficult, but probably  easier than the fourth. Basically, you have less options for viewing the world and interacting with it. On a graph, you have two dimensions, and a location on the graph is represented by x and y, perpendicular to each other. Three dimensions adds z, perpendicular to x and y. A two d being would not be able to see z though, so a being from the third dimension would not necessarily be visible unless it passed through the origin of z, or the plane that the two dimensional being lives in. The two dimensional being would not be able to see the whole three dimensional being, but rather only what was passing through the origin of its z. We could visualize this as not seeing a hand, but rather only a cross section of a couple fingers.

My whole point in this is to allow us (or more honestly, me) to imagine seeing the fourth dimension. Basically, a four dimensional object or being would appear as an ever changing three dimensional shape, or if it stopped moving, a very confusing stationary three dimensional shape. Using the previous example, four dimensional beings may exist in vast numbers, but never bother to pass through our plane of existence.

Grade of the Road

                I always wondered about those strange signs, such as the ones you see on a New Mexico highway rolling over steep hills with a black semi-truck going down a black triangle, reading 6%. I never managed to piece it together, although I never gave it much thought. It is simple enough to look such things up on the internet, so why not. It turns out grade is a simple rise over run calculation. For example, if you go a horizontal distance on a mountain highway of 100 feet, and you rise 25 feet, then your calculation is 25 feet divided 100 feet, which is equal to 0.25 (the feet cancel). You then times this number by 100, getting 25% grade, percent being out of a hundred. This is exactly like calculating the slope of a line on a graph, using its rise over run, except grade is represented as a percentage rather than a fraction or whole number.

              You can do more using the method of calculating the grade of a road. The horizontal distance can be seen as the base of a right triangle, and the vertical distance as the height. This makes a right triangle, which gives you the ability to solve for many things. For instance, you can solve for distance traveled (The hypotenuse) using the Pythagorean Theorem, which using our example would be calculating out 25 squared plus 100 squared, which is equal to c squared, which comes out to be about 103 feet, or more precisely 25 times the square root 17. You can also solve for the angle of the road, using trigonometry. Tangent is a ratio used for finding out the side opposite an angle and an adjacent side in a right triangle. If we use this equation, we can solve for the angle the road is tilting, in this case Tan (tangent) of x° is equal to 25 feet divided by 100 feet. You can’t divide by tangent to find x, however, so you must use the Inverse Tangent to solve for the angle, or the Inverse Tangent of 25 feet divided by 100 feet (0.25), which is about 14. Therefore, the angle of the road is about 14°.  This could be very helpful for people to understand how steep the road they are driving on is, since a 25% grade would be difficult to see.

Response to Jamm'n Peaches blog, Err in the Direction of Kindness

                What really got me thinking about the Jamm’n peaches blog, Err in the Direction of Kindness, is the Boy Scouts and how they have changed some of their exclusive practices. I think this is absolutely wonderful because to me it seems like a great program for anyone to be involved in and I find it ridiculous that it would be exclusive to someone just for religion or sexual preference. I have personally wondered about this myself, not to any particular point but just because I enjoy wondering about these things. I myself have worked on the welcome center at Albuquerque Academy with the Boy Scouts, who I think highly of for their help and hard work. It was interesting though, because one day we were building one of the shade structures, and before we got working one of the Leaders of the Boy Scout group got everyone together to say a prayer.  So, there I was, an agnostic standing respectfully but sort of awkwardly in the midst of a group of about twenty people praying. This got me thinking about what this organization, the Boy scouts were originally indented to be and how they were now, and what they would think about working with me if they knew my beliefs. Of course, I am sure they are all good people, who would hold no prejudice against me whatsoever, and I certainly am not going to complain about them praying, but it was something to think about.