temporary storage

When we think of a tangent line of a point on a curve as only touching that curve at one point, while still having a fixed slope, we have already shed light on the nature of lines and the Euclidean grid: if you were to begin rotating the tangent line, at what point would it cease to be a tangent line and become a secant line? It would become a secant line the instant it began rotating, as lines are infinitely thin. This notion of infinitely thin lines, and infinitely small increments of time, is at the heart of calculus.

Why is this ever counter-intuitive to us? It has been argued that, hundreds of years ago, "math-denial" was a means for rejecting the idea of change by those in power. That may be true, but that strays more into the subject of history, really. I think the real reason is that, in a strict psychological sense, the notion of 'infinitely small' things like points and lines registers as an impossibility; there can not be 'something' without substance; mass and volume. Intuitively, we think that in order for a line to exist, it must consist of 'something', and already have properties like length, width, mass, etc.. Because of this, 'points' are actually visualized/conceptualized as tiny circles/spheres - but a line overlapping a point can rotate and still be touching the point; which I think people then try to apply to spheres and curves, but in doing so find that it's incorrect. We are not born with the concept of 'hypothetical', it is learned.

Let's arrive at a technical conclusion concerning 'change' and frequency; the 'change' of 'change' in the real world. When you tighten the strings of a guitar or any stringed instrument to tune it, you do not have to tune the strings to a specific man-made 'note' of frequency to make the instrument playable, you merely have to make sure the strings of the instrument are 'in tune' with each other. Yet, based on frequency, we can use devices to find what frequency and 'note' a string is tuned to, and the frequency of a note does not change (allegedly). Using the example of strings, the string can range in frequency from 0 (being loose against the fretboard of a guitar) to infinity (although the string will obviously snap). Therefore, frequency does not change linearly, but exponentially. Let us then explain 'octave' notes in music (although the prefix oct-, meaning 8, is only correct with the 'chromatic' scale, which I'll get into momentarily):

frequency_octave.gif

Imagine that as we tune the string to a higher pitch/frequency, as the number of wave crests increases, the pitch of the instrument is in the process of cycling through 'notes' of music. When the top of the next crest comes to the same position in space (relative to the entire segment) which the top of the previous wave crest was at, the pitch reaches an 'octave note' (the pitch produced is recognizable as similar to the lower/higher octave note).

'Scales' in music divide the difference between a note and lower/higher octaves into notes - typically (and almost always), due to a system agreed upon hundreds of years ago called 'equal temperament', the frequency between octaves is divided into 12 notes, with the 1st and 12th notes being 'octave' notes.

An example of such a scale is the 'chromatic' scale (which is what uses the word 'octave' to refer to the note of recurring recognizability in frequency), which divides the frequency into a pattern of notes that anyone who has ever seen or played with a piano will be familiar with. The reason the recurring note is called an 'octave' note, is because in the chromatic scale, the 12 notes have 5 sharps/flats (one group of 2 and one group of 3), and otherwise consists of 7 distinguishable notes standard to the scale, with the 8th note repeating:

Going to jump ahead a bit here in the technical complexity of the topic, hope you guys don't mind:

Frequency and Entanglement

All particles that ever interact become 'entangled' to each other to a certain degree. In interacting, the frequency of their Zitterbewegung (trembling motion) has been synchronized with each other to a certain extent. The stronger the interaction, the greater the extent of synchronicity. The intensity of the synchronicity fades with time (if someone wants me to go back and give a more in depth explanation to this connection between frequency, 'interaction', and the entanglement of particles, I will).

Entanglement also extends to any particles the entangled particle interacts with; so any tunneling of electromagnetic radiation (such as down a wire, or in a computer processor) preserves this entanglement.