# What is Phi: Definition and 130 Discussions

Phi (; uppercase Φ, lowercase φ or ϕ; Ancient Greek: ϕεῖ pheî [pʰé͜e]; Modern Greek: φι fi [fi]) is the 21st letter of the Greek alphabet.
In Archaic and Classical Greek (c. 9th century BC to 4th century BC), it represented an aspirated voiceless bilabial plosive ([pʰ]), which was the origin of its usual romanization as ⟨ph⟩. During the later part of Classical Antiquity, in Koine Greek (c. 4th century BC to 4th century AD), its pronunciation shifted to that of a voiceless bilabial fricative ([ɸ]), and by the Byzantine Greek period (c. 4th century AD to 15th century AD) it developed its modern pronunciation as a voiceless labiodental fricative ([f]).
The romanization of the Modern Greek phoneme is therefore usually ⟨f⟩.
It may be that phi originated as the letter qoppa, and initially represented the sound /kʷʰ/ before shifting to Classical Greek [pʰ]. In traditional Greek numerals, phi has a value of 500 (φʹ) or 500,000 (͵φ). The Cyrillic letter Ef (Ф, ф) descends from phi.
As with other Greek letters, lowercase phi (encoded as the Unicode character U+03C6 φ GREEK SMALL LETTER PHI) is used as a mathematical or scientific symbol. Some uses, such as the golden ratio, require the old-fashioned 'closed' glyph, which is separately encoded as the Unicode character U+03D5 ϕ GREEK PHI SYMBOL.

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1. ### Rotate Points on Sphere by Theta and Phi

Dear Forum, My goal is to rotate several points on a sphere by a theta and phi. For example, I have a sphere where the elevation is theta (90 to -90) and the azimuthal is phi (-180 to 180). I have the following points on the sphere: theta = [45 45 45 45] phi = [-180 90 90 180] This generate...

30. ### Qubit Math: Partitioning Phi into BetaA and BetaB

Hi All, I'm gradually teaching myself quantum computing, being an excellent programmer and fair at linear algebra and geometry. To learn it, I'm writing a quantum computer emulator (complete with graphical representations). I'll share it when I get to some level of completion. The following...
31. ### Yaw Pitch & Roll to spherical Theta & Phi

hi guys, can someone please tell me how to find Theta and Phi from Yaw Pitch and Roll? I use my smartphone orientation sensor on a project and i need to calculate the smartphone's normal vector projection.
32. ### Name of something that is forced to look a certain way

I was looking up information about phi and how it seems apparent everywhere in nature, debatable. I stumbled upon this photo which is quite funny: This got me thinking about how people try to fit and force data to match whichever hypothesis they have. This is more jokingly, I think. However...
33. ### How to find phi in simple harmonic motion

Given a position vs time graph of simple harmonic motion of an object and using the equation x(t) = xo = A sin (ωt +Φ), how am I supposed to find Φ? I can easily find A and the period (T). I also know T = 2 pi / ω, so I can find ω. But I don't know x(t) and t. I tried finding random points in...
34. ### How Do You Calculate the Phase Constant φ from a Wave Displacement Graph?

1. The figure shows the displacement y versus time t of the point on a string at x = 0, as a wave passes through that point. The scale of the x axis is set by ys = 18.0 mm.The wave has form y(x, t) = ym sin (kx - ωt+φ). What is φ...
35. ### Apparent indiscrepancy in the average value of phi function

So, while solving a problem a friend came up with involving the Totient function, I ended up doing a bit of research into the average asymptotics of the function. On page 268 of Introduction to the Theory of Numbers, it's mentioned that "The average of order of ##\phi\left(n\right)## is...
36. ### Simple faraday's law problem (phi = what?)

Homework Statement A toroid with N windings and radius R has cross sectional radius x (x<<R). The current running through the wires is given by ##I = I_0\sin (\omega t)##. There is a magnetic field in the center of the toroid. A loop of wire of radius three times that of the...
37. ### MHB Tensor Products - D&F page 369 Example 3 - The map phi

I am reading Dummit and Foote, Section 10.4: Tensor Products of Modules. I am currently studying Example 3 on page 369 (see attachment). Example 3 on page 369 reads as follows: (see attachment) ------------------------------------------------------------------------------- In general...
38. ### Derivation of Phi-Hat wrt Phi in Spherical Unit Vectors

Homework Statement I just want to know how to get from this: ∂ø^/∂ø = -x^cosø - y^sinø to this: = -(r^sinθ+θ^cosθ) Homework Equations All the equations found here in the Spherical Coordinates section: http://en.wikipedia.org/wiki/Unit_vector The Attempt at a Solution I've...
39. ### Abstract algebra, show that phi is a homomorphism

Homework Statement The Attempt at a Solution I'm very new to this kind of maths, so don't quite know how to get started. If I understood the question at all we have g_i \mapsto \phi_i and so I have a homomorphism if I can show that \pi(g \cdot g_i) = \pi(g) \circ \pi(g_i) I'm thinking...
40. ### MHB Eulers phi function, orders, gcd

Let a, k , l , m e Z>1 and let a^k=1 (mod m) and a^l= 1 (mod m). Let d=gcd(k,l) Prove that a^d=1 (mod m). I get already confused at the start: Is it true that k|phi(m) (Lagrange) but k can also be a multiple of the order of a (mod m) and then it can be the other way round. Can anybody clarify...
41. ### How Is the Euler Totient Function Applied in Real Life?

What is most motivating way of introducing this function? Does it in itself have any real life applications that have an impact. I can only think of a^phi(n)=1 (mod n) which is powerful result but is this function used elsewhere.
42. ### MHB Exploring the Practical Applications of the Euler Totient Function

What is most motivating and tangible way of introducing this function? Does it in itself have any real life applications that have an impact. I can only think of a^phi(n)=1 (mod n) which is powerful result but is this function used elsewhere.
43. ### Finding angles of Phi. Very challenging.

1) Given: two ropes of negligible weight suspend an object weighing 50N from the ceiling There are two angles of PHI that form from as a measure from the vertical. PHI1 + PHI2= 60°. The tension of the left rope is given to be 80N and the tension of the left rope is given to be 70N. The entire...
44. ### Vector Calculus Proof: Curl V = 0 -> V = grad phi

If ∇ x v = 0 in all of three dimensional space, show that there exists a scalar function ##\phi (x,y,z)## such that v = ∇##\phi##. (from Walter Strauss' Partial Differential Equations, 2nd edition; problem 11; pg 20.) I'm not really sure where to begin with this problem. I asked a few of my...
45. ### Is Phi the Secret to Perceiving Beauty in Human Faces?

This whole post is just for fun -- here is an article about applying phi (Golden ratio) to human faces and esthetics of human beauty. http://www.goldennumber.net/face/ If you don't know much about Phi try:: http://en.wikipedia.org/wiki/Golden_ratio The authors constructed 1.6... x 1...
46. ### Find the potential fucntion, phi

Homework Statement Consider the vector ﬁeld given by F(x, y, z) = yz \hat{i} + xz \hat{j} + (xy + 3z^{2})\hat{k} a. Calculate ∇xF and show that F is a conservative field. Done, result = <0,0,0> which implies the vector field is conservative. b. The way we were taught this is to set...
47. ### What happens when a neutral pion decays?

The neutral phi meson decays to make two photons after around 8 x10-17 seconds Is this just straight annihilation of the quark anti quark pair? If it is this does it take this amount of time due the quark colour, as in do you need a red up and a red anti up for annihilation? Or does this...
48. ### What is the Proof for the Summation of Integers using Euler's Phi Function?

Homework Statement Prove: \sum_{k<n} k= \frac{n \phi (n)}{2} gcd(k,n)=1 \phi is Euler's phi function or Euler's totient function The Attempt at a Solution So the sum should be 1+2+3+...+(n-2)+(n-1) which will equal \frac{n^2-n}{2} using the formula for the sum of integers by...