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)

Each point in the circle circunscribed to a triangle give us a line (Wallace-Simson line)

Interactive 'Mostration' of the Wallace-Simson line.

All the Wallace-Simson lines form a deltoid (Steiner Deltoid).

Steiner deltoid is a hypocycloid related with the nine point circle of a triangle.

Steiner Deltoid and the Morley triangle are related.

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.

Central angle in a circle is twice the angle inscribed in the circle.

Interactive 'Mostation' of the property of central and inscribed angles in a circle.

A Dilative Rotation is a combination of a rotation an a dilatation from the same point.

He studied transformations of images, for example, faces.

In this painting we can see, among lots of interesting things, an anamorphosis of a skull. (In Spanish)

In an equiangular spiral the angle between the position vector and the tangent is constant.

Two transformations of an equiangular spiral with the same general efect.

A golden rectangle is made of an square and another golden rectangle.

A golden rectangle is made of an square an another golden rectangle. These rectangles are related through an dilative rotation.

Two equiangular spirals contains all vertices of golden rectangles.

The golden spiral is a good approximation of an equiangular spiral.

The volume of a tetrahedron is one third of the prism that contains it.

Special sections of a tetrahedron are rectangles (and even squares)

Howard Eves, mathematician and historian of Mathematics, received the George Polya Award for the article Two Surprising Theorems on Cavalieri Congruence

We want to study a surprising Cavalieri congruence between a sphere and a tetrahedron. In this page we can see sections in a sphere.

We show a sphere and the Howard Eves's tetrahedron with congruent sections.

One eighth of a regular dodecahedon of edge 2 has the same volume as a dodecahedron of edge 1.

One eighth of a regular dodecahedon of edge 2 has the same volume as a dodecahedron of edge 1.

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 twelve vertices of an icosahedron lie in three golden rectangles. Then we can calculate the volume of an icosahedron

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.

These polyhedra pack together to fill space, forming a 3 dimensional space tessellation or tilling.

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.

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.

Geometric sequences graphic representations

One intuitive example of how to sum a geometric series. A geometric series of ratio less than 1 is convergent.

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, the Euler's constant, is defined using a covergent series.

Power with natural exponents are simple and important functions.

We extend the definition of power of positive integer exponent and their inversas if we consider positive rational exponents.

We can see how the graph of a polynomial of degree three with three real roots changes when we change a root.

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.

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.

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.

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

Studying the volume of a barrel, Kepler solved a problem about maxima in 1615.

By increasing the degree, Taylor polynomial approximates the exponential function more and more.

Mercator published his famous series for the Logarithm Function in 1668. Euler discovered a practical series to calculate.

By increasing the degree, Taylor polynomial approximates the sine function more and more.

The function is not defined for values less than -1. Taylor polynomials about the origin approximates the function between -1 and 1.

The function has a singularity at -1. Taylor polynomials about the origin approximates the function between -1 and 1.

The function has a singularity at -1. Taylor polynomials about the origin approximates the function between -1 and 1.

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.

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.

Mercator published his famous series for the Logarithm Function in 1668. Euler discovered a practical series to calculate.

We can see it as a dilatative rotation.

The multiplication by a complex number is a transformation of the complex plane: dilative rotation.

From a complex number we can obtain a geometric progression obtaining the powers of natural exponent (multiplying successively)

Complex power functions with natural exponent have a zero (or root) of multiplicity n in the origin.

A complex polinomial of degree 3 has three zeros.

Every complex polynomial of degree n has n zeros or roots.

Every complex polynomial of degree n has n zeros or roots.

Representación de los óvalos de Cassini y la lemniscata.

Podemos modificar las multiplicidades del cero y del polo de estas funciones sencillas.

Una primera aproximación a estas transformaciones. Representación de dos haces coaxiales de circunferencias ortogonales.

The Complex Exponential Function extends the Real Exponential Function to the complex plane.

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 maps horizontal lines to confocal ellipses.

Inversion is a plane transformation that transform straight lines and circles in straight lines and circles.

Inversion preserves the magnitud of angles but the sense is reversed. Orthogonal circles are mapped into orthogonal circles

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.

We will see how Taylor polynomials approximate the function inside its circle of convergence.

The complex exponential function is periodic. His power series converges everywhere in the complex plane.

The power series of the Cosine Function converges everywhere in the complex plane.

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 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.

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 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.

It may be interesting to familiarize ourselves with the probabilities correspondig to different intervals in normal distributions.

T Student distribution was studied by William Gosset(1876-1937)

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 show us in 'The Method' how to use the lever law to discover the area of a parabolic segment.

Kepler used an intuitive infinitesimal approach to calculate the area of a circle.

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 was one mathematician who contributed to the origin of integral calculus. He used infinitesimal techniques for calculating areas and volumes.

Studying the volume of a barrel, Kepler solved a problem about maxima in 1615.

Using Cavalieri's Principle we can calculate the volume of a sphere.

Mercator published his famous series for the Logarithm Function in 1668. Euler discovered a practical series to calculate.