Visual Power Rules
Visual Log Rules
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Last edited by 1ucid; 01-06-15 at 20:05.
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Visualizing sine and cosine waves from a circe.
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Take a smooth, closed convex curve and slide a chord of constant length around it. Meanwhile consider a point on the moving chord that divides it into parts of length a and b. This point also traces out a closed curve as the chord makes a round.
What’s the area between the curves? By Holditch’s theorem, it is simply: π a b.
Amazing, right?
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Pitagorina teorema :
Prvo jedna slika (nadam se da nije "off topic" jer nije animacija
Dokaz preko "površine crnog" ... klik klak klik klak ...
I naravno, jedan "vodootporni" (waterproof) dokaz
Zanimljivo da nemoraju biti površine kvadrata nad stranicama . Umjesto kvadrata mogu biti površine bilo kakvih "matematički proporcionalno sličnih" (isti oblik - različite velicine) geometriskih slika
Naprimjer :
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Evo ne znam koliko se uklapam u temu, ali se radi o fraktalima:
Na moj zahtjev lik napravio ovu animaciju, opisuje vizuelni prikaz sledece jednakosti
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The Golden Ratio and the Great Pyramid of Giza
A square pyramid is a pyramid that has a square base. A regular square pyramid can be described by its apothem (a), semi-base (b), and height (h). When the ratio of these lengths (b:h:a) are equal to 1: √ ϕ: ϕ, the pyramid is known as a “golden pyramid”. (ϕ, or phi, is a symbol for the golden ratio, a ratio whose value is an irrational number with interesting properties.)
While most ancient Egyptian pyramids do not have this ratio, there is one pyramid that is remarkably close–The Great Pyramid of Giza. Some scientific historians argue that this accuracy was a mere accident and that ancient Egyptians were not familiar with phi or calculating such a specific slant height. Others disagree and believe that the ancient Egyptians who built the Great Pyramid of Giza truly did have knowledge of the golden ratio and used it. Source.
What do you think?
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The proof of Sum of Square Numbers!
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Last edited by 1ucid; 05-07-15 at 13:25.
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Coin rotation paradox
Consider two round coins of equal size. Imagine holding one still and rolling the other coin around it, making sure that it doesn’t slip. How many times will the outer coin rotate during a full revolution around the stationary coin?
If you (like most people) believe the answer to be only once, you’re wrong:
So the outer coin will already have made a full rotation when it reaches the opposite side, perhaps contrary to our intuition.
https://en.wikipedia.org/wiki/Coin_rotation_paradox
https://en.wikipedia.org/wiki/Cardioid
Related paradoxical effects involving rolling circles are Aristotle’s paradox and Copernicus’ theorem.
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Start with a circle-shaped piece of paper and mark one interior point. Now make a few creases by folding the circle’s boundary onto the marked point. Repeat this until you’ve made enough creases.
The envelope of these straight lines is a perfect ellipse, whose foci are the one marked point and the center of the original ellipse.
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Visualizing the area of a circle
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The following is a simple activity that uses playing cards for learning a simple number theory concept. This procedure is based from the book “Time Travel and Other Mathematical Bewilderments” by Martin Gardner.
First, remove all the cards of one suit (e.g. all the spades) from a pack of playing cards. Afterwards, arrange the cards face down in a row in an ascending numerical order (ace = 1, Jack = 11, Queen = 12 and King = 13), from left to right. Now, do the following steps.
Turn each of the card over.
Turn over every second card (2, 4, 6, etc.).
Turn over every third card (3, 6, 9, 12), then every fourth card and continue in this fashion until you have turned over the last card (the thirteenth card in this case).
After the procedure, all the cards except for the ace, four and nine would be face down.
Do you see a pattern here? Yes, all the numerical values of the cards that are face up are squares.
You might think that this is just a coincidence but it’s not. You can even try to use 30 blank cards and number them from 1 to 30 and do the procedure described above. After turning over the 30th card, you will see that the card number 1, 4, 9, 16 and 25 are the only cards face up.
Why Does This Happen?
I told you earlier that this activity will teach you a simple number theorem, right? The concept that this activity shows is that every positive integer has an even number of divisors (the divisors include 1 and the number itself) except for square numbers. For example, the divisors of number 10 are 1, 2, 5 and 10, a total of four divisors. Even every prime number has an even number of divisors since it is divisible by 1 and by itself. This means that all non-square numbers have an even number of divisors while square numbers have an odd number of divisors.
So, how was the activity related to the concept? When you are turning over the cards, the cards are turned over equivalent to the number of their numerical values’ divisors. For example, the 7 of Spades was turned two times because 7 has two divisors, 1 and 7. Since all the cards were face down in the beginning, when you turned the cards an even number of times, the end result is that the cards would be face down. However, if you turned the cards an odd number of times (e.g. the four of Spades was turned three times since 4 has three divisors, 1, 2, and 4), the cards would end up being face up. Hence, all the remaining cards that are face up are the square numbers 1, 4 and 9.
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A geometric construction of the parabola. The blue point is called the focus, and the horizontal line is the directrix. The blue lines show all the points which are at an equal distance from the red point and the blue focus point. The point of the blue line directly above the red dot contributes to the parabolic curve, because the parabola is defined as the set of all points equidistant to the focus and the directrix.
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