Geometry

Real Number Set Diagram

Beautiful proofs (#4) - When Gauss was a young child...

The legend goes something like this:

Using this ingenious method used by Gauss allows us to write a generic formula for the sum of first n positive integers as follows:

fractal nature

“a fractal object is an object of which each element is also a fractal object”

Greek Alphabet

Α α = alpha

Β β = beta

Δ δ = delta

Ε ε = epsilon

Φ φ = phi

Γ γ = gamma

Η η = eta

Ι ι = iota

Ξ ξ = ksi

Κ κ = kappa

Λ λ = lambda

Μ μ = mu

Ν ν = nu

Ο ο = omicron

Π π = pi

Ρ ρ = rho

Σ σ = sigma

Τ τ = tau

Θ θ = theta

Ω ω = omega

Χ χ = chi

Υ υ = upsilon

Ζ ζ = zeta

Ψ ψ = psi

Sphere is circumscribed by a cube and has a smaller cube inscribed within it. What is the ratio of the cube volumes?

Stanislaw Lem, Fiasco (via eka-mark)

A sample solution in the Lorenz attractor when ρ = 28, σ = 10, and β = 8/3. Source. 

In 1847 Oliver Byrne published an edition of Euclid's Elements which replaced text with pictures. Shown here is the statement that “in a right angled triangle the square on the hypotenuse is equal to the sum of the squares of the sides”.

In 1637, Pierre Fermat, a French lawyer, wrote a conjecture in the margin of an old Greek mathematical book: “It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second, into two like powers”. The conjecture was finally proved over three centuries later by Andrew Wiles.

Light reflecting off a curved mirror in a smoky room will generally create one of these three shapes, called the ‘swallowtail’, the ‘pyramid’, and the ‘purse’.

Mathematical Spirals

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. 

Hyperboloid

Archimedean Polyhedra

Here are the possibilities as to what can appear at a vertex. The notation (3, 4, 3, 4) means each vertex contains a triangle, a square, a triangle, and a square, in that cyclic order. 

These solids were described by Archimedes, although his original writings on the topic are  lost and only known of second-hand. All but one of these polyhedra were gradually rediscovered during the Renaissance by various artists, and Kepler finally reconstructed the entire set in 1619. A key characteristic of the Archimedean solids is that each face is a regular polygon, and around every vertex, the same polygons appear in the same sequence, e.g., hexagon-hexagon-triangle in the truncated tetrahedron, shown above. Two or more different polygons appear in each of the Archimedean solids, unlike the Platonic solids which each contain only a single type of polygon. The polyhedron is required to be convex. 

There are 13 solids classified as Archimedean, (not counting two mirror images twice). The snub cube and snub dodecahedron are chiral; they each come in two handednesses (two enantiomorphs, referred to as left-handed and right-handed forms or laevo and dextro forms). To see the enantiomorph of either of the snub figures, view the reflection of your computer screen in a mirror. Or, look at this side-by-side model of a snub cube and its enantiomorph. 

Although the accepted polyhedron names are less than ideal, there is a certain logic to the names above. (They are adapted from Kepler’s Latin terminology.) The term snub refers to a process of surrounding each polygon with a border of triangles as a way of deriving for example the snub cube from the cube. The term truncated refers to the process of cutting off corners. Compare, for example, the cube and the truncated cube. Truncation adds a new face for each previously existing vertex, and replaces n-sided polygons with 2n-sided ones, e.g., octagons instead of squares.  

There is also another class of polyhedra in which the same regular polygons appear at each vertex: the prisms and antiprisms, which have vertex types (4, 4, n) and (3, 3, 3, n) respectively. But as these form two infinite series, they can not be all listed, and so are described as a separate group. (Georgehart.com/Polyhedra/Archimedean)

Will Artificial Intelligence Surpass Our Own? - Scientific American (via wildcat2030)