Phi - More Facts and Figures
What is the connection between euler's number, phi (golden ratio) and pi () ? The relation between biological and mathematical top height in Scots pine. This relationship was derived after Oberg noticed an interesting relationship between pi and phi while contemplating geometric questions. do those constants have any relation to each other? does something like pi-e or pi/e has any significance?.
E, pi, phi | Physics Forums
It seems to have some amazing properties. If you draw a circle with a diameter of 1 meter, one sixth of the circumference will be equal to 1 cubit. Keep in mind that we weren't using the meter as a unit of measurement until some time after Originally intended to be one ten-millionth of the distance from the Earth's equator to the North Pole at sea levelits definition has been periodically refined to reflect growing knowledge of metrology.
The Great Pyramid is a 'square circle' as they say. This is another highly debated subject. Many people refuse to believe the Egyptians had knowledge of Pi or encoded it into their buildings. So what exactly is Pi?
Consequently, its decimal representation never ends and never repeats. Pi - Wikipedia Further more, the Great Pyramid has another very important number hidden within its geometry. If you take the surface area of the four top sides and divide it by the surface of the base, you'll get the 'golden number', also called the 'golden ratio'.
So just what is this golden number? In mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. That's the first quick way.
The second answer was spotted by Scott Beach but it is also in the table at the foot of this page: What is the surface area S of the brick? Above we saw that the sum of the 2 smaller face's areas equals the largest face's area, and that this is Phi. Since there are 2 faces with smallest area, 2 of middle-sized area which total 2 times the largest face area, that is 2 Phi and we also have two other faces of the largest area Phithen: The surface area of the brick is 4 Phi How long is the diagonal across the brick?
Another surprise awaits us when we calculate the length of the diagonal across the brick. The formula is a 3-dimensional analogue of Pythagoras Theorem.
- Golden ratio
- π/Φ Pi in the Phi
- Phi's Fascinating Figures
So how long is the diagonal of our Phi-bonacci brick? Since its sides x,y and z are 1, Phi and phi, the length of its diagonal is: I'll leave you to check the algebra but the surprisingly simple answer is The diagonal of the brick has length 2 A relationship between Phi and Pi Even more surprising is that the brick shows a simple relationship between Pi and Phi, using the values for the diagonal D that we have just found and the surface area S.
If we imagine the brick tightly packed into a sphere, the centre of the sphere will be half way along diagonal D. So the radius of the sphere will be 1 and its surface area will be 4 Pi.
We showed above that the surface area of the brick, S, is 4 Phi. Putting these two together we have: The ratio of the surface areas of the Phibonacci brick and its surrounding sphere is Phi: Pi You do the maths Suppose we have a brick with sides a, b and c. Suppose also that the sum of the smaller sides equals the longest side i.
Golden ratio - Wikipedia
The questions here show that the only way to have BOTH properties is when a and b as well as b and c are in the golden ratio Phi: The Golden rectangle and powers of phi On a golden rectangle with sides of length phi and 1, dividing at the golden ratio point gives two overlapping squares whose sides are phi. The gap between a square and the longer side has length 1-phi which is also phi2. So the phi by phi squares are themselves divided at their own golden section ratios by the side of the other overlapping square.
Higher powers of phi Here is another geometrical illustration of phi2, phi3 and phi4: If we take a square with sides of length 1 and divide two sides at the golden section point, we form two new squares yellow and blue and two red rectangles.Sacred Geometry 101F (Part 2): Pi - Phi - Fibonacci Sequence
Notice that the diagonal is also divided at its golden section point too. The large yellow square has sides of length phi, so its area is phi2. Evaluate Phi and raise it to the power 4 hint: Save this value in a memory or write it down. Write Phi6 in the same form. Can you spot a general pattern for Phin?
Here is a cube with sides divided at the golden section points to split it into eight pieces.
The eight pieces are of 4 different shapes: There are 8 blocks in the cube: What is the volume of each? Each volume will be a power of phi!
E, pi, phi
If the total volume is 1 since the original cube has sides of length 1what relationship does this suggest between your multiples of powers of phi and 1? How is this answer related to the relationship you found in the last question? A relation on Powers of Phi There's an even more intriguing relationship between the powers of Phi than the one you discovered for yourself in the last section.
Here's how you can find it: What do you notice about the difference between Phi3 and Phi2?