Algebra is the offer made by the devil to the mathematician. The devil says: I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvelous machine.Sir Michael Atiyah (via curiosamathematica)
dover mathematics paperbacks (1953, 1956-1959 eds.)
I absolutely love Dover publishing. I have so many books from them that are filled with beautiful mathematics.
(via visualizingmath)
- Quora Question: How do I explain recursion to a four-year-old?
- Answerer: Explain it to someone a year younger than you and ask them to do the same.
A drawing by the 19th century artist J.M.W. Turner, showing how a sphere can cast a shadow on a slanted plane in the shape of an ellipse.
(via visualizingmath)
According to Wikipedia, a spiral is a curve which emanates from a central point, getting progressively farther away as it revolves around the point (similar to helices [plural for helix!] which are three-dimensional). Pictured above are some of the most important spirals of mathematics.
Logarithmic Spiral: Equation: r=ae^bθ. I must admit that these are my favorite! Logarithmic spirals are self-similar, basically meaning that the spiral maintains the same shape even as it grows. There are many examples of approximate logarithmic spirals in nature: the spiral arms of galaxies, the shape of nautilus shells, the approach of an insect to a light source, and more. Additionally, the awesome Mandelbrot set features some logarithmic spirals. Fun fact: the Fibonacci spiral is an approximation of the Golden spiral which is only a special case of the Logarithmic spiral.
Fermat’s Spiral: Equation: r= ±θ^(½). This is a type of Archimedean spiral and is also known as the parabolic spiral. Fermat’s spiral plays a role in disk phyllotaxis (the arrangement of leaves in a plant system).
Archimedean Spiral: Equation: r=a+bθ. The Archimedean spiral has the property that the distance between each successive turning of the spiral remains constant. This kind of spiral can have two arms (like in the Fermat’s spiral image), but pictured above is the one-armed version.
Hyperbolic Spiral: Equation: r=a/θ. It is also know as the reciprocal spiral and is the opposite of an Archimedian spiral. It begins at an infinite distance from the pole in the center (for θ starting from zero r = a/θ starts from infinity), and it winds faster and faster around as it approaches the pole; the distance from any point to the pole, following the curve, is infinite.
I had learned to do integrals by various methods shown in a book that my high school physics teacher Mr. Bader had given me. [It] showed how to differentiate parameters under the integral sign – it’s a certain operation. It turns out that’s not taught very much in the universities; they don’t emphasize it. But I caught on how to use that method, and I used that one damn tool again and again. [If] guys at MIT or Princeton had trouble doing a certain integral, [then] I come along and try differentiating under the integral sign, and often it worked. So I got a great reputation for doing integrals, only because my box of tools was different from everybody else’s, and they had tried all their tools on it before giving the problem to me.
Richard Feynman [2, pp. 71–72]
Differentiating Under the Integral Sign - PDF
(via olena)
(Source: math.uconn.edu, via olena)
What do you see? Some kind of band spinning around, or something twisting?
The motion of the first band shown above is ambiguous. Its more obvious that the band on the left is spinning, whereas the band on the right is twisting. The particular dynamics are made evident by watching the two black points on the bands edges. But without these defining features the motion of the gray band is indistinguishable between the two cases.
Perhaps the band is not moving at all, and instead its just the observer’s perspective rotating around this twisted yet inert band!
Can you think of any other shape that might exhibit this kind of symmetry?
Inspired by Bees&Bombs
Mathematica code:
x[ r_, w_, s_, t_] :=
{r (1 + w*Cos[2 Pi*t]) Cos[2 Pi *s],
r (1 + w*Cos[2 Pi*t]) Sin[2 Pi*s] ,
w*Sin[ 2 Pi*t]}
P[ Q_, N_, r_, w_, f_, S_, T_] :=
Table[x[r, w, -s + n/N, f*t + q/Q + f*n/N] ,
{s, 1/S, 1, 1/S}, {t, 1/T, 1, 1/T}, {q, 0, Q - 1, 1}, {n, 0, N, 1 }]
V = P[2, 200, 1, .25, 1, 50, 50];
Manipulate[
Graphics3D[{
{{Thickness[.001], Line[V[[s, t]]]},
Table[
{FaceForm[GrayLevel[.5]], Opacity[1], EdgeForm[None],
Polygon[
{V[[s, t, 1, n]], V[[s, t, 2, n]], V[[s, t, 2, n + 1]],
V[[s, t, 1, n + 1]], V[[s, t, 1, n]]}]},
{n, 1, 200, d}]},
{PointSize[p], Black, Point[{V[[s, t, 1, 1]], V[[s, t, 2, 1]]}]}},
Background -> White, Lighting -> "Neutral", PlotRange -> 3.5,
ViewPoint -> .4 {1, 1, .6}, Boxed -> False, ImageSize -> 500],
{t, 1, 50, 1}, {s, 1, 50, 1}, {d, 1, 5, 1}, {p, {0, .015}}]