Ian Beardsley

About me

Ian Beardsley is author of the books, "Cosmic Archeology" and "Cosmic Archeology: Knowing the Universe by Looking at Ourselves", he is also a flamenco guitarist. He has created this blog to explore basic concepts he feels are important as we enter the space age. See his math sytem below under "My Work".

My work

Formulas Derived from the Parallelogram

Remarks. Squares and rectangles are parallelograms that have four sides the same length, or two sides the same length. We can determine area by measuring it either in unit triangles or unit squares. Both are fine because they both are equal sided, equal angled geometries that tessellate. With unit triangles, the areas of the regular polygons that tessellate have whole number areas. Unit squares are usually chosen to measure area.



Having chosen the unit square with which to measure area, we notice that the area of a rectangle is base times height because the rows determine the amount of columns and the columns determine the amount of rows. Thus for a rectangle we have:



A=bh



Drawing in the diagonal of a rectangle we create two right triangles, that by symmetry are congruent. Each right triangle therefore occupies half the area, and from the above formula we conclude that the area of a right triangle is one half base times height:



A=(1/2)bh



By drawing in the altitude of a triangle, we make two right triangles and applying the above formula we find that it holds for all triangles in general.



We draw a regular hexagon, or any regular polygon, and draw in all of its radii, thus breaking it up into congruent triangles. We draw in the apothem of each triangle, and using our formula for the area of triangles we find that its area is one half apothem times perimeter, where the perimeter is the sum of its sides:



A=(1/2)ap



A circle is a regular polygon with an infinite amount of infitesimal sides. If the sides of a regular polygon are increased indefinitely, the apothem becomes the radius of a circle, and the perimeter becomes the circumference of a circle. Replace a with r, the radius, and p with c, the circumference, and we have the formula for the area of a circle:



A=(1/2)rc



We define the ratio of the circumference of a circle to its diameter as pi. That is pi=c/D. Since the diameter is twice the radius, pi=c/2r. Therefore c=2(pi)r and the equation for the area of a circle becomes:



A=(pi)r^2



(More derived from the parallelogram)



Divide rectangles into four quadrants, and show that



A. (x+a)(x+b)=(x^2)+(a+b)x+ab

B. (x+a)(x+a)=(x^2)+2ax+(a^2)

A. Gives us a way to factor quadratic expressions.

B. Gives us a way to solve quadratic equations. (Notice that the last term is the square of one half the middle coefficient.)



Remember that a square is a special case of a rectangle.



There are four interesting squares to complete.



1) The area of a rectangle is 100. The length is equal to 5 more than the width multiplied by 3. Calculate the width and the length.

2) Solve the general expression for a quadratic equation, a(x^2)+bx+c=0

3) Find the golden ratio, a/b, such that a/b=b/c and a=b+c.

4) The position of a particle is given by x=vt+(1/2)at^2. Find t.



Show that for a right triangle (a^2)=(b^2)+(c^2) where a is the hypotenuse, b and c are legs. It can be done by inscribing a square in a square such that four right triangles are made.



Use the Pythagorean theorem to show that the equation of a circle centered at the origin is given by r^2=x^2+y^2 where r is the radius of the circle and x and y the orthogonal coordinates.



Derive the equation of a straight line: y=mx+b by defining the slope of the line as the change in vertical distance per change in horizontal distance.



Triangles


All polygons can be broken up into triangles. Because of that we can use triangles to determine the area of any polygon.


Theorems Branch 1

1. If in a triangle a line is drawn parallel to the base, then the lines on both sides of the line are proportional.

2. From (1) we can prove that: If two triangles are mutually equiangular, they are similar.

3. From (2) we can prove that: If in a right triangle a perpendicular is drawn from the base to the right angle, then the two triangles on either side of the perpendicular, are similar to one another and to the whole.

4. From (3) we can prove the Pythagorean theorem.

Theorems Branch 2

1. Draw two intersecting lines and show that opposite angles are equal.

2. Draw two parallel lines with one intersecting both. Use the fact that opposite angles are equal to show that alternate interior angles are equal.

3. Inscribe a triangle in two parallel lines such that its base is part of one of the lines and the apex meets with the other. Use the fact that alternate interior angles are equal to show that the sum of the angles in a triangle are two right angles, or 180 degrees.

Theorems Branch 3

1. Any triangle can be solved given two sides and the included angle.

c^2=a^2+b^2-2abcos(C)

2. Given two angles and a side of a triangle, the other two sides can be found.

a/sin(A)=b/sin(B)=c/sin(C)

3.Given two sides and the included angle of a triangle you can find its area, K.

K=(1/2)bc(sin(A))

4.Given three sides of a triangle, the area can be found by using the formulas in (1) and (3).

Question: what do parallelograms and triangles have in common?
Answer: They can both be used to add vectors.

Trigonometry

When a line bisects another so as to form two equal angles on either side, the angles are called right angles. It is customary to divide a circle into 360 equal units called degrees, so that a right angle, one fourth of the way around a circle, is 90 degrees. The angle in radians is the intercepted arc of the circle, divided by its radius, from which we see that in the unit circle 360 degrees is 2(pi)radians, and we can relate degrees to radians as follows:

Degrees/180 degrees=Radians/pi radians

An angle is merely the measure of separation between two lines that meet at a point.

The trigonometric functions are defined as follows:

cos x=side adjacent/hypotenuse

sin x=side opposite/hypotenuse

tan x=side opposite/side adjacent



csc x=1/sin x

sec x=1/cos x

cot x=1/tan x

We consider the square and the triangle, and find with them we can determine the trigonometric function of some important angles.

Square (draw in the diagonal): cos 45 degrees =1/sqrt(2)=sqrt(2)/2

Equilateral triangle (draw in the altitude): cos 30 degrees=sqrt(3)/2; cos 60 degrees=1/2

Using the above formula for converting degrees to radians and vice versa:



30 degrees=(pi)/6 radians; 60 degrees=(pi)/3 radians.

The regular hexagon and pi

Tessellating equilateral triangles we find we can make a regular hexagon, which also tessellates. Making a regular hexagon like this we find two sides of an equilateral triangle make radii of the regular hexagon, and the remaining side of the equilateral triangle makes a side of the regular hexagon. All of the sides of an equilateral triangle being the same, we can conclude that the regular hexagon has its sides equal in length to its radii. If we inscribe a regular hexagon in a circle, we notice its perimeter is nearly the same as that of the circle, and its radius is the same as that of the circle. If we consider a unit regular hexagon, that is, one with side lengths of one, then its perimeter is six, and its radius is one. Its diameter is therefore two, and six divided by two is three. This is close to the value of pi, clearly, by looking at a regular hexagon inscribed in a circle.

The sum of the angles in a polygon

Draw a polygon. It need not be regular and can have any number of sides. Draw in the radii. The sum of the angles at the center is four right angles, or 360 degrees. The sum of the angles of all the triangles formed by the sides of the polygon and the radii taken together are the number of sides, n, of the polygon times two right angles, or 180 degrees. The sum of the angles of the polygon are that of the triangles minus the angles at its center, or A, the sum of the angles of the polygon equals n(180 degrees)-360 degrees, or

A=180 degrees(n-2)

With a rectangular coordinate system you need only two numbers to specify a point, but with a triangular coordinate system --- three axes separated by 120 degrees -- you need three. However, a triangular coordinates system makes use of only 3 directions, whereas a rectangular one makes use of 4.

A rectangular coordinate system is optimal in that it can specify a point in the plane with the fewest numbers, and a triangular coordinate system is optimal in that it can specify a point in the plane with the fewest directions for its axes. The rectangular coordinate system is determined by a square, and the triangular coordinate system by an equilateral triangle. They are the basis for many mosaics in Moorish castles, such as those in the Alhambra in Spain.

From the Physics Notebook of Ian Beardsley
F=ma M=mv v=x/t
F=Force M=momentum m=mass v=velocity x=distance t=time a=acceleration
M=mv=m(x/t) a=dv/dt a(dt)=dv v=at v=dx/dt dx/dt=at dx=at(dt) x=(1/2)at^2
x=x_0+vt+(1/2)at^2
int[x^n] 0 to x =(x^(n+1))/(n+1) and (d/dx)x^n=nx^(n-1)
K=kinetic energy U=potential energy C=constant
K=(1/2)mv^2 U=mgy h=height
K+U=C mgh=mgy+(1/2)mv^2 or (1/2)m(v_0)^2=U+K where v_0=initial velocity
Work=W=Fx and U=-W
Thus work is the distance traveled or moved by the component of the force in that direction, and potential energy is the negative of the work. Use the definition for work and the chain rule for derivatives to show that kinetic energy (energy of motion) is as given above. The chain rule is:
dv/dt=(dv/dx)(dx/dt)
A ball rolling on an incline will stay in motion until it attains the same height on another incline facing the first, even if the
inclinations of the two inclines are not the same. If there is no second incline, the ball will never attain the original height and will therefore continue to roll forever, unless otherwise acted on by a force, like friction. For every force there is an equal but opposite reaction.
Notice that:
mgh=(1/2)m(v_0)^2