matematicas visuales visual math
Geometry

Triangles
Morley triangle | matematicas visuales
If we trisect the angles of a triangle and consider the three intersection points of corresponding trisector lines we always get an equilateral triangle (Morley's triangle)
Wallace-Simson lines | matematicas visuales
Each point in the circle circunscribed to a triangle give us a line (Wallace-Simson line)
Wallace-Simson lines | Mostration | matematicas visuales
Interactive 'Mostration' of the Wallace-Simson line.
Steiner deltoid | matematicas visuales
All the Wallace-Simson lines form a deltoid (Steiner Deltoid).
Steiner deltoid is an hypocycloid | matematicas visuales
Steiner deltoid is a hypocycloid related with the nine point circle of a triangle.
The deltoid and the Morley triangle | matematicas visuales
Steiner Deltoid and the Morley triangle are related.
Pythagoras' Theorem in a tiling | matematicas visuales
We can see Pythagoras' Theorem in a tiling. It is a graphic demonstration of Pythagoras' Theorem we can see in some floor made using squares of two different sizes.

Circles
Central and inscribed angles in a circle | matematicas visuales
Central angle in a circle is twice the angle inscribed in the circle.
Central and inscribed angles in a circle | Mostration | matematicas visuales
Interactive 'Mostation' of the property of central and inscribed angles in a circle.

Plane Transformations
Dilative rotation | matematicas visuales
A Dilative Rotation is a combination of a rotation an a dilatation from the same point.
Durer | matematicas visuales
He studied transformations of images, for example, faces.
Los Embajadores de Holbein el Joven | matematicas visuales
In this painting we can see, among lots of interesting things, an anamorphosis of a skull. (In Spanish)

Spirals
Equiangular spiral | matematicas visuales
In an equiangular spiral the angle between the position vector and the tangent is constant.
Dilation and rotation in an equiangular spiral | matematicas visuales
Two transformations of an equiangular spiral with the same general efect.

The Golden Proportion
The golden rectangle | matematicas visuales
A golden rectangle is made of an square and another golden rectangle.
The golden rectangle and the dilative rotation | matematicas visuales
A golden rectangle is made of an square an another golden rectangle. These rectangles are related through an dilative rotation.
The golden rectangle and two equiangular spirals | matematicas visuales
Two equiangular spirals contains all vertices of golden rectangles.
The golden spiral | matematicas visuales
The golden spiral is a good approximation of an equiangular spiral.

Space Geometry
The volume of the tetrahedron | matematicas visuales
The volume of a tetrahedron is one third of the prism that contains it.
Sections on a tetrahedron | matematicas visuales
Special sections of a tetrahedron are rectangles (and even squares)
Sections in Howard Eves's tetrahedron | matematicas visuales
Howard Eves, mathematician and historian of Mathematics, received the George Polya Award for the article Two Surprising Theorems on Cavalieri Congruence
Sections in the sphere | matematicas visuales
We want to study a surprising Cavalieri congruence between a sphere and a tetrahedron. In this page we can see sections in a sphere.
Surprising Cavalieri congruence between a sphere and a tetrahedronn | matematicas visuales
We show a sphere and the Howard Eves's tetrahedron with congruent sections.
Regular dodecahedron | matematicas visuales
One eighth of a regular dodecahedon of edge 2 has the same volume as a dodecahedron of edge 1.
Volume of a regular dodecahedron (Flash version) | matematicas visuales
One eighth of a regular dodecahedon of edge 2 has the same volume as a dodecahedron of edge 1.
Volume of an octahedron | matematicas visuales
The volume of an octahedron is four times the volume of a tetrahedron. It is easy to calculate and then we can get the volume of a tetrahedron.
The icosahedron and its volume | matematicas visuales
The twelve vertices of an icosahedron lie in three golden rectangles. Then we can calculate the volume of an icosahedron
The volume of a truncated octahedron | matematicas visuales
The truncated octahedron is an Archimedean solid. It has 8 regular hexagonal faces and 6 square faces. Its volume can be calculated knowing the volume of an octahedron.
The truncated octahedron is a space-filling polyhedron | matematicas visuales
These polyhedra pack together to fill space, forming a 3 dimensional space tessellation or tilling.
Hexagonal section of a cube | matematicas visuales
We can cut in half a cube by a plane and get a section that is a regular hexagon. Using eight of this pieces we can made a truncated octahedron.
A truncated octahedron made by eight half cubes | matematicas visuales
Using eight half cubes we can make a truncated octahedron. The cube tesselate the space an so do the truncated octahedron. We can calculate the volume of a truncated octahedron.

Real Analysis

Sequences and Series
Geometric sequence | matematicas visuales
Geometric sequences graphic representations
Sum of a geometric series of ratio 1/4 | matematicas visuales
One intuitive example of how to sum a geometric series. A geometric series of ratio less than 1 is convergent.
Sum of a geometric series of ratio 1/2 | matematicas visuales
The geometric series of ratio 1/2 is convergent. We can represent this series using a rectangle and cut it in half successively. Here we use a rectangle such us all rectangles are similar.
Gamma, Euler's constant | matematicas visuales
Gamma, the Euler's constant, is defined using a covergent series.

Powers and Polynomials
Powers with natural exponents | matematicas visuales
Power with natural exponents are simple and important functions.
Powers with positive rational exponents | matematicas visuales
We extend the definition of power of positive integer exponent and their inversas if we consider positive rational exponents.
Polynomial of degree 3 with 3 real roots | matematicas visuales
We can see how the graph of a polynomial of degree three with three real roots changes when we change a root.
Polynomial approximations | matematicas visuales
The powers of natural exponent are the base of the polynomials. We can consider the polynomic function that passes through a series of points of the plane. This is an interpolation problem that is solved here using the Lagrange interpolating polynomial.

Integral
Riemann integral | matematicas visuales
The integral concept is associate to the concept of area. We began considering the area limited by the graph of a function and the x-axis between two vertical lines.
The error approximating the integral | matematicas visuales
When approximating an integral using rectangles we commit an error. In some cases, for example, in monotonic functions, we can limit the magnitude of the error.
Integral of powers | matematicas visuales
Archimedes' Method to calculate the area of a parabolic segment | matematicas visuales
Archimedes show us in 'The Method' how to use the lever law to discover the area of a parabolic segment.
Kepler: The best proportions for a wine barrel | matematicas visuales
Studying the volume of a barrel, Kepler solved a problem about maxima in 1615.

Taylor's Polynomials
Taylor polynomials: Exponential function | matematicas visuales
By increasing the degree, Taylor polynomial approximates the exponential function more and more.
Mercator and Euler: Logarithm Function | matematicas visuales
Mercator published his famous series for the Logarithm Function in 1668. Euler discovered a practical series to calculate.
Taylor polynomials: Sine function | matematicas visuales
By increasing the degree, Taylor polynomial approximates the sine function more and more.
Taylor polynomials: Square root | matematicas visuales
The function is not defined for values less than -1. Taylor polynomials about the origin approximates the function between -1 and 1.
Taylor polynomials: Rational function 1 | matematicas visuales
The function has a singularity at -1. Taylor polynomials about the origin approximates the function between -1 and 1.
Taylor polynomials: Rational function 2 | matematicas visuales
The function has a singularity at -1. Taylor polynomials about the origin approximates the function between -1 and 1.
Taylor polynomials: Rational function with two real singularities | matematicas visuales
This function has two real singularities at -1 and 1. Taylor polynomials approximate the function in an interval centered at the center of the series. Its radius is the distance to the nearest singularity.
Taylor polynomials: Rational function without real singularities | matematicas visuales
This is a continuos function and has no real singularities. However, the Taylor series approximates the function only in an interval. To understand this behavior we should consider a complex function.

Exponentials and Logarithms
Una propiedad de la integral de la hipérbola (Spanish) | matematicas visuales
Una propiedad de la integral de la hipérbola | 'Mostración'(Spanish) | matematicas visuales
El logaritmo de un producto (Spanish) | matematicas visuales
Definición de logaritmo como una integral (Spanish) | matematicas visuales
Mercator and Euler: Logarithm Function | matematicas visuales
Mercator published his famous series for the Logarithm Function in 1668. Euler discovered a practical series to calculate.
Approximation of number e (Spanish) | matematicas visuales
Two definitions of number e (Spanish) | matematicas visuales
The exponential as the inverse of the logarithm (Spanish) | matematicas visuales
Hipérbolas, logaritmos y exponenciales (Spanish) | matematicas visuales
Exponencial functions (Spanish) | matematicas visuales
Radioactiv desintegration (Spanish) | matematicas visuales

Complex Analysis

Product of complex numbers
Multiplying two complex numbers | matematicas visuales
We can see it as a dilatative rotation.
The product as a complex plane transformation | matematicas visuales
The multiplication by a complex number is a transformation of the complex plane: dilative rotation.
Geometric sequence | matematicas visuales
From a complex number we can obtain a geometric progression obtaining the powers of natural exponent (multiplying successively)

Complex Functions
Powers with natural exponent | matematicas visuales
Complex power functions with natural exponent have a zero (or root) of multiplicity n in the origin.
Polynomial of degree 3 | matematicas visuales
A complex polinomial of degree 3 has three zeros.
Polynomial of degree n | matematicas visuales
Every complex polynomial of degree n has n zeros or roots.
Polynomial of degree n (variant) (Spanish) | matematicas visuales
Every complex polynomial of degree n has n zeros or roots.
Polynomial of degree 2 (Spanish) | matematicas visuales
Representación de los óvalos de Cassini y la lemniscata.
Cero and polo (Spanish) | matematicas visuales
Podemos modificar las multiplicidades del cero y del polo de estas funciones sencillas.
Cero and polo (variant) (Spanish) | matematicas visuales
Moebius transformations (Spanish) | matematicas visuales
Una primera aproximación a estas transformaciones. Representación de dos haces coaxiales de circunferencias ortogonales.
The Complex Exponential Function | matematicas visuales
The Complex Exponential Function extends the Real Exponential Function to the complex plane.
The Complex Cosine Function | matematicas visuales
The Complex Cosine Function extends the Real Cosine Function to the complex plane. It is a periodic function that shares several properties with his real ancestor.
The Complex Cosine Function: mapping an horizontal line | matematicas visuales
The Complex Cosine Function maps horizontal lines to confocal ellipses.
Inversion | matematicas visuales
Inversion is a plane transformation that transform straight lines and circles in straight lines and circles.
Inversion: an anticonformal transformation | matematicas visuales
Inversion preserves the magnitud of angles but the sense is reversed. Orthogonal circles are mapped into orthogonal circles
Multifunctions: Powers with fractional exponent | matematicas visuales
The usual definition of a function is restrictive. We may broaden the definition of a function to allow f(z) to have many differente values for a single value of z. In this case f is called a many-valued function or a multifunction.

Taylor's Polynomials
Taylor polynomials: Rational function with two complex singularities | matematicas visuales
We will see how Taylor polynomials approximate the function inside its circle of convergence.
Taylor polynomials: Complex Exponential Function | matematicas visuales
The complex exponential function is periodic. His power series converges everywhere in the complex plane.
Taylor polynomials: Complex Cosine Function | matematicas visuales
The power series of the Cosine Function converges everywhere in the complex plane.

Probability

Random Variables
Binomial distribution | matematicas visuales
When modeling a situation where there are n independent trials with a constant probability p of success in each test we use a binomial distribution.
Poisson distribution | matematicas visuales
Poisson distribution is discrete (like the binomial) because the values that can take the random variable are natural numbers, although in the Poisson distribution all the possible cases are theoretically infinite.
Normal distribution | matematicas visuales
The Normal distribution was studied by Gauss. This is a continuous random variable (the variable can take any real value). The density function is shaped like a bell.
One, two and three standar deviations | matematicas visuales
One important property of normal distributions is that if we consider intervals centered on the mean and a certain extent proportional to the standard deviation, the probability of these intervals is constant regardless of the mean and standard deviation of the normal distribution considered.
Calculating probabilities in Normal distributions | matematicas visuales
It may be interesting to familiarize ourselves with the probabilities correspondig to different intervals in normal distributions.
T Student distribution (Spanish) | matematicas visuales
T Student distribution was studied by William Gosset(1876-1937)
Calculating probabilities in t Student distributions (Spanish) | matematicas visuales

History

Pitagoras
Pythagoras' Theorem in a tiling | matematicas visuales
We can see Pythagoras' Theorem in a tiling. It is a graphic demonstration of Pythagoras' Theorem we can see in some floor made using squares of two different sizes.

Archimedes
Archimedes' Method to calculate the area of a parabolic segment | matematicas visuales
Archimedes show us in 'The Method' how to use the lever law to discover the area of a parabolic segment.

Kepler
Kepler: The Area of a Circle | matematicas visuales
Kepler used an intuitive infinitesimal approach to calculate the area of a circle.
Kepler: Surface and volume of a sphere | matematicas visuales
Kepler studied the volume and surface of the sphere. He thought the volume of the sphere as made up of small cones, then he sum all of these cones and get a relation between the surface of a sphere en its volume.
Kepler: The volume of a wine barrel | matematicas visuales
Kepler was one mathematician who contributed to the origin of integral calculus. He used infinitesimal techniques for calculating areas and volumes.
Kepler: The best proportions for a wine barrel | matematicas visuales
Studying the volume of a barrel, Kepler solved a problem about maxima in 1615.

Cavalieri
Cavalieri: The volume of a sphere | matematicas visuales
Using Cavalieri's Principle we can calculate the volume of a sphere.

The Logarithm Function
Mercator and Euler: Logarithm Function | matematicas visuales
Mercator published his famous series for the Logarithm Function in 1668. Euler discovered a practical series to calculate.