spring-of-mathematics

spring-of-mathematics:

Infinity …    
      … it’s not big …
      … it’s not huge …
      … it’s not tremendously large …
      … it’s not extremely humongously enormous …
      … it’s

       …Endless!

Infinity has no end. Infinity is the idea of something that has no end.

"Paul Erdős lived in Budapest, Hungary, with his Mama. Mama loved Paul to infinity ∞. When Paul was 3. She had to go back to work as a math teacher….” (Extract from the book: The Boy Who Loved Math: The Improbable Life of Paul Erdős by Deborah Heiligman - Figure 1).

Infinity, most often denoted as infty(symbol:∞), is an unbounded quantity that is greater than every real number, is an abstract concept describing something without any limit and is relevant in a number of fields, predominantly mathematics and physics. In number systems incorporating infinitesimals, the reciprocal of an infinitesimal is an infinite number, i.e., a number greater than any real number. Infinity is a very tricky concept to work with, as evidenced by some of the counterintuitive results that follow from Georg Cantor’s treatment of infinite sets.
Georg Cantor formalized many ideas related to infinity and infinite sets during the late 19th and early 20th centuries. In the theory he developed, there are infinite sets of different sizes (called cardinalities). For example, the set of integers is countably infinite, while the infinite set of real numbers is uncountable. (Here is one of Proofs)

  • In Geometry and topology: Main article: Dimension (vector space). Infinite-dimensional spaces are widely used in geometry and topology, particularly as classifying spaces, notably Eilenberg−MacLane spaces. Common examples are the infinite-dimensional complex projective space K(Z,2) and the infinite-dimensional real projective space K(Z/2Z,1).
  • In Fractal Geometry: The structure of a fractal object is reiterated in its magnifications. Fractals can be magnified indefinitely without losing their structure and becoming “smooth”; they have infinite perimeters—some with infinite, and others with finite surface areas. One such fractal curve with an infinite perimeter and finite surface area is the Koch snowflake.
  • In Real analysis: In real analysis, the symbol \infty, called “infinity”, is used to denote an unbounded limit. x -> ∞ means that x grows without bound, and x  -> - ∞ means the value of x is decreasing without bound.

See more at: Infinity on Wikipedia and Mathworld - What is Infinity? on MathisFun.

Reference:  Paul Erdös and the Erdös Number Project page.

Image: Koch snowflakes & The Boy Who Loved Math: The Improbable Life of Paul Erdős.

spring-of-mathematics

spring-of-mathematics:

Type of Spirals: A spiral is a curve in the plane or in the space, which runs around a centre in a special way.
Different spirals follow. Most of them are produced by formulas:The radius r(t) and the angle t are proportional for the simplest spiral, the spiral of Archimedes. Therefore the equation is:
(3) Polar equation: r(t) = at [a is constant].
From this follows
(2) Parameter form:  x(t) = at cos(t), y(t) = at sin(t),
(1) Central equation:  x²+y² = a²[arc tan (y/x)]².

You can make a  spiral by two motions of a point: There is a uniform motion in a fixed direction and a motion in a circle with constant speed. Both motions start at the same point. 
(1) The uniform motion on the left moves a point to the right. - There are nine snapshots.
(2) The motion with a constant angular velocity moves the point on a spiral at the same time. - There is a point every 8th turn.
(3) A spiral as a curve comes, if you draw the point at every turn(Image).

Figure 1: (1) Archimedean spiral - (2) Equiangular Spiral (Logarithmic Spiral, Bernoulli’s Spiral).
Figure 2 : (1) Clothoide (Cornu Spiral) - (2) Golden spiral (Fibonacci number).

More Spirals: If you replace the term r(t)=at of the Archimedean spiral by other terms, you get a number of new spirals. There are six spirals, which you can describe with the functions f(x)=x^a [a=2,1/2,-1/2,-1] and  f(x)=exp(x), f(x)=ln(x). You distinguish two groups depending on how the parameter t grows from 0.

Figure 4:  If the absolute modulus of a function r(t) is increasing, the spirals run from inside to outside and go above all limits. The spiral 1 is called parabolic spiral or Fermat’s spiral.
Figure 5: If the absolute modulus of a function r(t) is decreasing, the spirals run from outside to inside. They generally run to the centre, but they don’t reach it. There is a pole.  Spiral 2 is called the Lituus (crooked staff).

Figure 7: Spirals Made of Line Segments.

Source:  Spirals by Jürgen Köller.

See more on Wikipedia:  SpiralArchimedean spiralCornu spiralFermat’s spiralHyperbolic spiralLituus, Logarithmic spiral
Fibonacci spiral, Golden spiral, Rhumb line, Ulam spiral
Hermann Heights Monument, Hermannsdenkmal.

Image: I shared at Spirals by Jürgen Köller - Ferns by Margaret Oomen & Ferns by Rocky.

spring-of-mathematics

spring-of-mathematics:

Mathematics and Traditional Cuisine

The mathematics of Pasta: A process analysis to find unity, formulas and ways to express structure mathematics of pasta shapes, by their mathematical and geometric properties.
See more at: The Maths of Pasta by George L. Legendre.

Image: 

  • 'Pasta By Design' - Created by a team of designers, ‘Pasta by Design’ book reveals the hidden mathematical beauty of pasta: its geometrical shapes and surfaces are explained by mathematical formulae, drawings and illustrations.
  • Animated gifs - From video: The traditional pasta making techniques used at Della Terra Pasta by Chris Becker [Video] - shared at here.

Types of Pasta in the post (From left to right):  Agnolotti - Tortellini - Saccottini - Sagne Incannulate - Pappardelle.

fouriestseries
fouriestseries:

Taylor Series Approximations
A Taylor series is a way to represent a function in terms of polynomials. Since polynomials are usually much easier to work with than complicated functions, Taylor series have numerous applications in both math and physics.
There are many equations in physics — like the one describing the motion of a pendulum — that are impossible to solve in terms of elementary functions. “Approximations using the first few terms of a Taylor series can make [these] otherwise unsolvable problems” solvable for a restricted area of interest [1].
The GIF above shows the five-term Taylor series approximation of a sine wave about x=0.
Mathematica code:
f[x_] := Sin[x]
ts[x_, a_, nmax_] := 
    Sum[(Derivative[n][f][a]/n!)*(x - a)^n, {n, 0, nmax}]
Manipulate[Plot[{f[x], ts[x, 0, nmax]}, {x, -2*Pi, 2*Pi}, 
    PlotRange -> {-1.45, 1.45}, 
    PlotStyle -> {{Thick, Cyan}, {Thick, Dotted, Yellow}}, 
    AxesStyle -> LightGray, Background -> Darker[Gray, 0.8]], 
    {nmax, 1, 30, 1}]

fouriestseries:

Taylor Series Approximations

A Taylor series is a way to represent a function in terms of polynomialsSince polynomials are usually much easier to work with than complicated functions, Taylor series have numerous applications in both math and physics.

There are many equations in physics — like the one describing the motion of a pendulum — that are impossible to solve in terms of elementary functions. “Approximations using the first few terms of a Taylor series can make [these] otherwise unsolvable problems” solvable for a restricted area of interest [1].

The GIF above shows the five-term Taylor series approximation of a sine wave about x=0.

Mathematica code:

f[x_] := Sin[x]
ts[x_, a_, nmax_] := 
    Sum[(Derivative[n][f][a]/n!)*(x - a)^n, {n, 0, nmax}]
Manipulate[Plot[{f[x], ts[x, 0, nmax]}, {x, -2*Pi, 2*Pi}, 
    PlotRange -> {-1.45, 1.45}, 
    PlotStyle -> {{Thick, Cyan}, {Thick, Dotted, Yellow}}, 
    AxesStyle -> LightGray, Background -> Darker[Gray, 0.8]], 
    {nmax, 1, 30, 1}]

「集団的自衛権の行使容認に賛成ですか?」と聞いたら、

反対が多いのに、

「限定的に行使容認はどうですか?」と聞いたら、

賛成が多くなるという。

馬鹿もたいがいにしろ!

「入れていいか?」と聞かれたら、「いやだ」と言うが、

「先っぽだけ」と言ったら、「それなら」とOKする。

そういう馬鹿女と一緒だよ。

まったく大衆は馬鹿だ。

こんな馬鹿な大衆が日本国の「立憲主義」の崩壊に

気付きもしないで、サッカーに夢中で日本の勝利を祈ってる。

日本が負けても渋谷の交差点でハイタッチして騒ぎまくる。

限定的に容認する馬鹿に誇りは持てない https://www.gosen-dojo.com/index.php?key=joqv0has1-1998#_1998