Polar And Parametric Curves

Here are some screenshots I uploaded from my TI-89 Titanium calculator via a USB Silverlink Cable through the TI Connect program that can be download on the Texas Instruments Website :) Update 3/22/2011 - Here are some screenshots I've compiled of a 3o1i Cycloid using parametric matrix equations for [X(t),Y(t)]...

(click image to enlarge)

If you like this content: Donate With Paypal

Dot Configuration Pyramid Spiral

 
If you like this content: Donate With Paypal

Texas Hold-Em Lingo

If you like this content: Donate With Paypal

Dice Probabilities

If you like this content: Donate With Paypal

Fourier Analysis and ODEs

Fourier Anylisis (where w = 2pi/T): Specific Fourier Waveforms: 0th and 1st order differential equations: 2nd order differential equations: Tailor approximations for input functions: nth order linear ODE's and transfer functions: Rules for Laplace Transforms: Laplace Transforms for specific functions: Special differential equation forms:
If you like this content: Donate With Paypal

The Morality Scale

If you like this content: Donate With Paypal

Circuit Resonance

The following shows a Simple Resonant Circuit (SRC) on PSpice:
Here the SRC is supergenerated by an operational amp then boosted by a transistor amplifier producing tones out a speaker:


If you like this content: Donate With Paypal

Fractal Dimensions

Spacetime generated from Fractals: If you like this content: Donate With Paypal

Special Relativity

What happens if a ball is traveling at a universally constant speed, say measured to be 50 mph straight up relative to the people on a bus thats moving 40 mph relative to a sidewalk, but also measured at 50 mph at a 3-4-5 triangle relative to the people on the sidewalk?
In that case the ball exhibits a "special relativity" as it travels at a constant speed relative to all observers, which means either time is technically dilated or the bus length is technically shortened:
This phenominon of Special Relativity time dilation has been shown to exist for the decay times of particles in atom blasters:

If you like this content: Donate With Paypal

Gravitational Collisions

Here is a representation of elastic and inelastic collisions using an experimental air track system and collision magnets: Here are the initial steps for reducing a 2 dimensional elastic collision down to a 1 dimensional elastic collision: Here is an example problem of the additional steps for solving a 1 dimensional elastic collision: Here is my own depiction of the dynamic gravitational forces on a rocket performing a planetary flyby: This shows the way in which a gravitational interaction can be modeled similar to a collision with the center of mass at the more massive body (gravitational collisions are a more complicated process as linear and angular momentum as well as gravitational kinetic energy may or may not be conserved all depending on the specific parameters, perhaps aerobraking or rocket propulsion are taken into account, and other effects such as magnetic forces and light or plasma radiation play an effect too depending on the circumstances): Here is an artist's depiction of the longterm gravitational trajectories that asteroids tend to take as they interact with planets and moons:

If you like this content: Donate With Paypal

Q-BYTE

A Q-BYTE is a quantum byte that exists in a superposition of all states simultaneously:

Notice all the reflection and rotation symmetries: Highly useful in MATRIX Programming Languages.



If you like this content: Donate With Paypal

Spiral Fractals

Spiral Fractals generated via Iterating Functions in the Complex Plane:

If you like this content: Donate With Paypal

Turbulent Flow Fields

If you like this content: Donate With Paypal

Ellipses and Unity

(1) The red line on the ellipse is the same length where it is touching a point anywhere on the ellipse. The endpoints of that line are called the foci of the ellipse.

(2) The diameter of the larger circle is exactly twice the diameter of the inner circle that is rotating around inside it.

(3) The diameters of the dynamic circles that are spinning around the outside of the stationary circles is equal to (3a)the stationary circle diameter, (3b)half the stationary circle diameter, (3c) a third of the stationary circle diameter, and (3d) a fourth of the stationary circle diameter, respectively. The flower pattern is called an epicycloid.

(4) This shows that the circumference of a circle, when unrolled is equivalent to pi times the diameter.

(5) This shows that as the number of vertexes on a perfect shape increases, that shape conforms more and more to the shape of a circle. It also shows the roots of unity along the imaginary unit circle, where the formula for the roots of unity: X(x,n) = 1^(1/n) = exp(x*2*Pi*i/n) = cos(x*2*Pi/n) + i*sin(x*2*Pi/n) where:

X(x,n)=> the xth root of the n roots of unity;

x=> the root number (going from 1 to n);

n=> the number of roots;

(Note: All Gif images are a courtessy of Wolfram's Mathworld which is the best resource for studying math around next to getting a personal tutor.)

If you like this content: Donate With Paypal