Mathematics & Physics
Mathematics Introduction
T
o me, Mathematics is like a secret code to the universe. I have enjoyed it ever since I was a kid when I first saw the just-released Walt Disney film,
Donald in Mathmagic Land. In the words of Galileo Galilei:
"Mathematics is the alphabet with which God has written the Universe"
Interesting Histories in Mathematics
A
lgebra is from the Arabic word,
al-jabr (reunion, restoration). It came from the title of a book,
Hibab al-jabr wal-muqubala (The Book of Reunion and Balancing), written in Baghdad in the year 825, by the Arab Mathematician,
Muhammad ibn Mūsā al-Khwārizmī (780-850). The word,
Algorithm (logical steps for solving a problem) is from the Latin translation of this book,
Algoritimi de numero Indorum. The book was a treatise on the Indian system of decimal numeration. He was the first to use
zero as a placeholder when using the
Hindu-Arabic Numerals. He also presented the first solution for Linear and Quadratic equations. Throughout the Renaissance period in Europe he was known as the
Inventor of Algebra. In addition to all of this, he was an accomplished astronomer and a geographer.
A
s a side note, the word
al-jabr was also used by the
Moors of Spain since they were of Arabic descent. To the Moors, an
Algebrista was a bonesetter, or "
restorer" of bones. Throughout Europe during medieval times, Barbers also called themselves an
Algebrista, since barbers often did bone-setting, bloodletting, and tooth extraction on the side. This gave rise to the red and white (
Blood &
Bone) striped barber poles in front of Barber shops. At the base of the pole was a
brass basin for collecting blood. When spinning, the
RED stripes gave the impression of blood flowing down to fill the basin. When NOT spinning, the
RED stripes represented the bloody bandages wraped around a patient's arm. Sometimes a
BLUE stripe was added that represented venous blood. The
BLUE stripe is now used in America to match the colors in the flag. Here I am focused on
Mathematics, but the
History of Barber Poles is a fascinating one, having to do with midieval bloodletting, dentistry, and even
current uses in prostitution in Asia. All of this evolving from the Arabic word for
reunion,
al-jabr.
T
rigonometry is from the Greek, trigónon (triangle) and metron (measure). Although some of the concepts were used as long ago as when the pyramids were built in ancient Eqypt, the field emerged with a renewed vigor during the 3rd and 4th centuries from studies related to astronomy.
C
alculus, from the Latin, calculus (pebble, counter), was created and developed in the 17th century by Sir Isaac Newton (1642-1726) and independently by Gottfried Wilhelm Leibniz (1646-1716) as a tool to better explain the Laws of Gravitation and Motion. In 1687 Sir Isaac Newton published his famous book, Philosophia Naturalis Principia Mathematics, which laid the foundation for all clasical mechanics.
O
f special mention is Leonardo Fibonacci (1170-1250), an Italian Mathematician (a.k.a. Leonardo of Pisa). He published a book in 1202 entitled Liber Abaci (Book of Calculation) which introduced Europe to Fibonacci Numbers & the Golden Ratio. He is credited with being the first to introduce Europe to the Hindu-Arabic decimal number system. Many consider him to be the most talented Western Mathematician of the middle ages.
The Golden Ratio, Phi (φ)
φ = 1.6180339887…
1/φ = .6180339887…
↑↑ Golden Rectangles create a Golden Spiral
The Pentagram Contains Many Golden Ratios (φ)
It can re-create itself indefinitely!
☞
☞
☞
☞
☞
A knot tied with paper will ALWAYS generate a pentagon.
Hover Over It to reveal the hidden Pentagram. ↓
Let Red = 1
then White = φ (1.6180339887…)
The Fibonacci Series → Golden Ratio (φ)
Pick ANY two numbers, NOT BOTH ZERO. (fractional, big, small, negative, whatever)
Create a Fibonacci Series by simply adding together the preceding two numbers to create the next.
Clasic Fibonacci Series: 1, 2, 3, 5, 8, 13, 21, 34, 55, …
A Random Fibonacci Series: -1.5, 16, 14.5, 30.5, 45, 70.5, 120.5, 191, 311.5, …
The RATIO of any term in the series to the previous term approaches the Golden Ratio (φ).
In math terms, lim_{n→∞} X_{n} ⁄ X_{n-1} = φ
The Best Number is 73
7 x 3 = 21 and 73 happens to be the 21^{st} prime number
7 and 3 are also primes
The mirror of 73 is 37, which is the 12^{th} prime (and 12 is the mirror of 21)
In Binary (Base 2):
3 = 11
7 = 111
21 = 10101
73 = 1001001
ALL are Palindromes (the same read backwards and forward)
Also, 73 (1001001) has 7 digits and 3 ones
In Octal (Base 8):
73 = 111
Also a Palindrome
73 is also a "Star Number" and a "Centered Figurate Number"
With a center Hexagon of 37 figures (the mirror of 73), we can get a Star of 73 figures!
The next Star Number is 121
121 contains both 12 & 21, which represent the 12^{th} and 21^{st} prime numbers, 37 and 73
(The Star Number 121 is used for the game of Chinese Checkers)
Explore more about the number 73 on Wikipedia
The Interesting Number 108
108 = 1^{1} x 2^{2} x 3^{3}
In Number Theory, 108 is an Abundant Number and a Semiperfect Number
As an Abundant Number we will add up it's divisors which are: 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, and 54.
When added together: 1+2+3+4+6+9+12+18+27+36+54 = 172 (which is 64 more Abundant than 108)
2 sin (108° ⁄ 2 ) = φ (The Golden Ratio)
108 = 9 dozen
In Christianity there are 108 beads in the Rosary
In Hinduism, Buddhism, and Jainism the number 108 is considered sacred
In Buddhist temples in Japan, a bell is chimed 108 times in to finish the old year and welcome in the new one. Each ring represents one of 108 earthly temptations a person must overcome to achieve nirvana.
In Buddhism there are 108 ⁄ 2 = 54 prayer beads in the Japa Mala
In most Buddhist Temples there are 108 steps representing the 108 questions asked of Buddha
In martial arts there are 108 pressure points on the human body
In the Yang (long) form of Tai Ji Juan there are 108 moves
At 108°F the human body's organs begin to fail
The Earth to the Sun distance is 108 times the diameter of the Sun
108 is the number of stitches in a Major League Baseball
In 2016 the Chicago Cubs finally won the World Series for the first time in 108 years.
Their win came in the 10th inning with 8 runs (108).
Explore more about the number 108 on Wikipedia
Algebra
The Quadratic Equation
ax² + bx + c = 0
The solution, where a ≠ 0, is …
x = -b ± √ b² - 4ac
2a
a.k.a. a
Second Degree Polynomial Equation
Below is a
Completing the Square solution using the following algorithm:
(x + h)² = x² + 2hx + h² where
h = ^{b}⁄_{2a} In step 4 below we add
h² to
Complete the Square
1. The Quadratic Equation
ax² + bx + c = 0
2. Divide by a
x² + ^{b}⁄_{a}x + ^{c}⁄_{a} = 0
3. Subtract ^{c}⁄_{a}
x² + ^{b}⁄_{a}x = - ^{c}⁄_{a}
4. Add ^{b²}⁄_{4a²}
x² + ^{b}⁄_{a}x + ^{b²}⁄_{4a²} = - ^{c}⁄_{a} + ^{b²}⁄_{4a²}
5. Write as a square
(x + ^{b}⁄_{2a})² = - ^{c}⁄_{a} + ^{b²}⁄_{4a²}
6. Clean up right side
(x + ^{b}⁄_{2a})² = ^{b²}⁄_{4a²} - ^{c}⁄_{a} = (b² - 4ac) ⁄_{4a²}
7. Take the square root
x + ^{b}⁄_{2a} = (± √b² - 4ac ) ⁄ _{2a}
8. Subtract ^{b}⁄_{2a}
x = (-b ± √b² - 4ac ) ⁄ _{2a}
Dividing Fractional Expressions
b ≠ 0 , c ≠ 0 , d ≠ 0
Exponents & Logarithms
Exponents
In its simplest form an Exponent is
the number of times one number (x) is repeated in a multiplication.
y = x · x · x · x · x …
Let a = how many x's there are in the multiplication, then we can write the above product as:
y = x^{ a}
We pronounce this as, " y equals x to the a "
If a = 2, " y equals x squared "
If a = 3, " y equals x cubed "
Here are some Exponent rules.
x^{ a} · x^{ b} = x^{ a+b}
x^{ a}÷ x^{ b} = x^{ (a-b)}
x^{ a} · y^{ a} = (xy)^{ a}
(x^{ a})^{ b} = x^{ (ab)}
x^{ -a} = ^{1}⁄ x^{ a}
x^{ (a ⁄ b)} = ^{b}√x^{ a}
x^{ 0} = 1
x^{ 1} = x
e^{ ix} = cos(x) + i·sin(x)
e^{ iπ} = -1 (Euler's Formula)
Logarithms
From the Greek logos 'ratio' and arithmos 'number'
Again, we consider:
y = x^{ a}
The Logarithm of number y is the Exponent a to which a base number x must be applied.
Logarithms are the Inverse Function to the Exponent.
In other words, we ask, " To get the number y, how many times (a) do we multiply the number (x)? "
y = x · x · x · x · x …?
Logarithm notation is written so:
a = log_{x}(y) where y = x^{ a}
We pronounce this as, " a is the log to the base x of y "
Here are some Logarithm rules.
When n = 2 it is called the Binary Logarithm and is written:
log_{2}(x) where x = 2^{ y}
When n = 10 it is called the Common Logarithm and is written:
log(x) where x = 10^{ y}
When n = e (Euler's Number) it is called the Natural Logarithm and is written:
ln(x) where x = e^{ y}
The two functions are mirrored about an x = y axis
e =
Lim
n → ∞
(1 +
)^{ n}
Euler's Number
e^{ x} is quite unique. The derivative (slope) of e^{ x} is the value of e^{ x}.
ƒ′ (e^{ x}) = e^{ x}
Trigonometry
The Setup
- First we draw a horizontal line ONE unit long.
- The line's left side endponit will be the Origin. As the line sweeps around counter-clockwise from this origin, it describes a circle of Radius = 1 and Circumference = 2π.
- As the line sweeps it describes an angle θ measured from the original horizontal line.
The Definitions
- Each angle α describes a right triangle whose hypotenuse, C = 1.
- The vertical line A = sin(α)
- The horizontal line B = cos(α)
- The ratio of the sin to the cos is the tangent. tan(α) = ^{A} ⁄ _{B}
sin α =
= cos β
cos α =
= sin β
tan α =
=
A² + B² = C²
The Pythagorean Theorem
Double Angle Formulas
sin (2A) = 2 sin A · cos A
cos (2A) = cos²(A) - sin²(A)
As the angle,
α, increases past
2π Radians (360°)
sin and
cos cycle between -1 and 1
and graph as a sine wave…
A wheel spinning along a straight path
also generates a sine wave.
This is known as
Simple Harmonic Motion…
Trigonometric Algebra
sin(α + β) = sin(α)·cos(β) ± cos(α)·sin(β)
cos(α + β) = cos(α)·cos(β) ∓ sin(α)·sin(β)
Trigonometric Series
Gottfried Wilhelm Leibniz developed these Series Formulae for Sine and Cosine
(x in radians) (! is factorial, e.g. 5! = 5 x 4 x 3 x 2 x 1)
sin x = x - (x^{3} ⁄_{3!}) + (x^{5} ⁄_{5!}) - (x^{7} ⁄_{7!}) + …
cos x = 1 - (x^{2} ⁄_{2!}) + (x^{4} ⁄_{4!}) - (x^{6} ⁄_{6!}) + …
Calculus
Differentiation (Tangent Slope)
The
Derivative of a function is defined as the
Tangent Slope at each x of the function
and is written as:
^{d}⁄_{dx} ƒ(x)
The derivative, with respect to x, of the function of x
A shorthand notation is:
ƒ′ (x)
f prime of x
^{d}⁄_{dx} x^{n} = n x^{(n-1)}
^{d}⁄_{dx} e^{x} = e^{x}
^{d}⁄_{dx} sin(x) = cos(x)
^{d}⁄_{dx} cos(x) = - sin(x)
Partial Differentiation (∂)
For functions with
multiple variables (x, y, t, …), the
Partial Derivative is defined as the
Derivative of the function with all variables EXCEPT ONE held constant. It is written as:
^{∂}⁄_{∂t} ƒ(x, y, t …)
The partial derivative, with respect to t, of the function of x, y, t etc.
Integrals (Sum of Areas Under Curve)
The Integral of a function is defined as the
Sum of the Areas between
the function and the x-axis and is written as:
∫ ƒ(x) dx
The integral of the function of x with respect to x
The
Definite Integral of a function is the
Sum of the Areas between
the function and the x-axis Only between x=a and x=b and is written as:
_{a} ∫^{ b}ƒ(x) dx
The integral, from a to b, of the function of x with respect to x
Some examples (n ≠ -1) …
The Bell Curve
S
tatistical Analysis makes use of families of distribution curves which are known as
Bell Curves. Here is one.
φ(x) = ke ^{(- π x²)}
Where
k = height of the Bell Curve
Curves of Constant Width
A
Curve of Constant Width is a special shape whose width (the distance between two parallel lines that each touch the shape's boundary) is a constant value. This means that it can
Fit and Rotate within a Square. I think these are very cool. The shape is used in many applications including the
Wankel Engine (which uses a Reuleaux Triangle),
British 20p and 50p coins, and the
Canadian 11-sided Dollar coin. Since the coins have a constant width they easily pass through the coin measuring mechanisms in vending machines. Circles of course, are a special case of these curves. Curves of constant width can easily be created from regular and irregular polygons with odd numbers of sides, like a triangle or a pentagon.
My Bermuda $3 Coin
The Amazing Ellipse
A
n
Ellipse is a conic section formed by a plane slicing through a cone at an angle. I think Ellipses are really cool. Ellipses have two focal points, from which the curve can be drawn. The orbit of any object around another describes an Ellipse. Every point on an Ellipse is the sum of the distance from the curve to the focal points. A special case of an Ellipse is the
Circle where
F_{1}=F_{2}. In an Elliptical Hall, sound waves will move outwards from one focus and concentrate at the other focus. The Ellipse formula is:
An ellipse can be defined from two fixed foci F_{1} , F_{2} using a given constant length L where:
L = (F_{1}, P) + (P, F_{2})
👨🏫
The Ellipse ~ Wikipedia
⚛
Elliptical Acoustics
Hyperphysics
The Hohmann Transfer Orbit
The Hohmann Transfer Orbit Ellipse
I
have always been fascinated by
Orbital Mechanics. Moving untethered through space with only the pure mathematical forces of the cosmos guiding you is amazing to me. Transferring from one circular orbit to another utilizes an
Ellipse with the center of
Mass as one foci. The process is named after
Walter Hohmann who first published it in 1925.
T
he Hohmann Transfer requires enough fuel for two ΔV 'burns'. ΔV_{1} enters the Elliptical Transfer Path. The spacecraft speed then 'bleeds off' along the path as it moves further away from the Gravitational Center of Mass M. As the spacecraft arrives at the Higher Orbit 2 and the Apogee of the Ellipse, another 'burn' ΔV_{2} is required to match the new Orbital Velocity. The TOTAL Energy requirement is:
Δ V = Δ v_{1} + Δ v_{2}
T
he process is 100% reversible, i.e. it can be used to go from a Lower Orbit to a Higher Orbit or vice versa.
v_{o} = √
v_{o} = Orbital Velocity
r = radius of orbit
Δ v_{1} = √
(
√
- 1
)
Δ v_{1} = Change in Velocity to enter Ellipse
Δ v_{2} = √
(
1 -
√
)
Δ v_{2} = Change in Velocity to enter Orbit 2
For Reference
Perigee = the nearest point from a Center of Mass in an orbit
Apogee = the farthest point from a Center of Mass in an orbit
Geostationary Orbit ~ The Orbital Velocity matches Earth's Rotional Velocity and is aligned directly over the Equator. These are always circular orbits at 35,786 km (22,236 mi) above the Earth's Equator and rotating in the same direction as Earth. The value R (distance from center of planet) = 42,164 km (26,200 mi).
GPS (Global Positioning System) Orbits are usually Geostationary Orbits. In order for them to be useful and accurate the signals sent and received MUST take into account the relativistic effects of the Speed of Light in a Gravitational Field.
Geosynchronous Orbit ~ The Orbital Velocity matches Earth's Rotional Velocity but these orbits are inclined to the Equator and can be quite elliptical.
Polar Orbits are Low Earth Orbits circling from pole to pole in about 1.5 hours. In 1 day they can see all of the Earth's surface as it rotates below. They are very useful for monitoring changes in the Earth's environment.
1 Earth Rotation, a.k.a. 1 Sidereal Day = 23 hours, 56 minutes, 4.09 seconds OR 23.93447 hours
🌎
Earth Orbit Speed
Calculator
📡
GPS ~ Global
Positioning System
📱
Hohmann Transfer
Orbit Calculator
Gravitational Assist (Fly By)
A Gravitational Assist Hyperbola
T
he 'Fly By' or
Gravitational Assist Trajectory is a
Hyperbolic Path whereby a spacecraft is allowed to accelerate towards a Mass but then just misses the Mass. This results in the spacecraft exiting from the encounter with greater speed and a changed direction, the so called
Slingshot effect. The maneuver is used routinely in space flight to maximize velocity, conserve fuel, and alter direction. It requires no fuel use in
ΔV 'burns'. I call it
Planetary Pinball.
O
f course any increased momentum imparted to the spacecraft must necessarily be given up by the planet. But since the planet's overall momentum is so enormous compared to the spacecraft's, the decrease in the planet's momentum (
Δρ = M Δv) is insignificant by comparison.
🚀
Gravity Assist Orbit
Voyager 2 Video
One Method to Construct a Pentagon
1. Draw circle A
2. Draw circle B (2x circle A)
3. Draw arc from bottom of circle B to near side of circle A
4. Draw arc from bottom of circle B to far side of circle A
5. Connect the 5 equal PTs