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. Case I: When the arc is half a circle the inscribed angle is a right angle.

Interactive 'Mostation' of the property of central and inscribed angles in a circle. Case II: When one chord that forms the inscribed angle is a diameter.

Interactive 'Mostation' of the property of central and inscribed angles in a circle. The general case is proved.

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.

The diagonal of a regular pentagon are in golden ratio to its sides and the point of intersection of two diagonals of a regular pentagon are said to divide each other in the golden ratio or 'in extreme and mean ratio'.

From Euclid's definition of the division of a segment into its extreme and mean ratio we introduce a property of golden rectangles and we deduce the equation and the value of the golden ratio.

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.

There is a standarization of the size of the paper that is called DIN A. Successive paper sizes in the series A1, A2, A3, A4, and so forth, are defined by halving the preceding paper size along the larger dimension.

Transforming a circle we can get an ellipse (as Archimedes did to calculate its area). From the equation of a circle we can deduce the equation of an ellipse.

Every ellipse has two foci and if we add the distance between a point on the ellipse and these two foci we get a constant.

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 calculate the surface area of sections of a sphere using the Pythagorean Theorem. We also study the relation with the Geometric Mean and the Right Triangle Altitude Theorem.

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.

A cuboctahedron is an Archimedean solid. It can be seen as made by cutting off the corners of a cube.

A cuboctahedron is an Archimedean solid. It can be seen as made by cutting off the corners of an octahedron.

The compound polyhedron of a cube and an octahedron is an stellated cuboctahedron.It is the same to say that the cuboctahedron is the solid common to the cube and the octahedron in this polyhedron.

The stellated octahedron was drawn by Leonardo for Luca Pacioli's book 'De Divina Proportione'. A hundred years later, Kepler named it stella octangula.

We study different prisms and we can see how they develop into a plane net. Then we explain how to calculate the lateral surface area.

Plane nets of prisms with a regular base with different side number cut by an oblique plane.

We study different cylinders and we can see how they develop into a plane. Then we explain how to calculate the lateral surface area.

We study different cylinders cut by an oblique plane. The section that we get is an ellipse.

Plane net of pyramids and pyramidal frustrum. How to calculate the lateral surface area.

Plane net of pyramids cut by an oblique plane.

Plane developments of cones and conical frustum. How to calculate the lateral surface area.

Plane developments of cones cut by an oblique plane. The section is an ellipse.

Several pages about simple techniques for building polyhedra: cardboard, origami, tubes, zome, tensegrity.

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.

Two points determine a stright line. As a function we call it a linear function. We can see the slope of a line and how we can get the equation of a line through two points. We study also the x-intercept and the y-intercept of a linear equation.

Power with natural exponents are simple and important functions. Their inverse functions are power with rational exponents (a radical or a nth root)

Polynomials of degree 2 are quadratic functions. Their graphs are parabolas. To find the x-intercepts we have to solve a quadratic equation. The vertex of a parabola is a maximum of minimum of the function.

Polynomials of degree 3 are cubic functions. A real cubic function always crosses the x-axis at least once.

We can consider the polynomial 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 derivative of a lineal function is a constant function.

The derivative of a quadratic function is a linear function, it is to say, a straight line.

The derivative of a cubic function is a quadratic function, a parabola.

Lagrange polynomials are polynomials that pases through n given points. We use Lagrange polynomials to explore a general polynomial function and its derivative.

If the derivative of F(x) is f(x), then we say that an indefinite integral of f(x) with respect to x is F(x). We also say that F is an antiderivative or a primitive function of f.

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.

If we consider the lower limit of integration a as fixed and if we can calculate the integral for different values of the upper limit of integration b then we can define a new function: an indefinite integral of f.

Monotonic functions in a closed interval are integrable. In these cases we can bound the error we make when approximating the integral using rectangles.

The integral of power functions was know by Cavalieri from n=1 to n=9. Fermat was able to solve this problem using geometric progressions.

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.

It is easy to calculate the area under a straight line. This is the first example of integration that allows us to understand the idea and to introduce several basic concepts: integral as area, limits of integration, positive and negative areas.

To calculate the area under a parabola is more difficult than to calculate the area under a linear function. We show how to approximate this area using rectangles and that the integral function of a polynomial of degree 2 is a polynomial of degree 3.

We can see some basic concepts about integration applied to a general polynomial function. Integral functions of polynomial functions are polynomial functions with one degree more than the original function.

The Fundamental Theorem of Calculus tell us that every continuous function has an antiderivative and shows how to construct one using the integral.

The Second Fundamental Theorem of Calculus is a powerful tool for evaluating definite integral (if we know an antiderivative of the function).

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 polynomial of degree 2 has two zeros or roots. In this representation you can see Cassini ovals and a lemniscate.

A complex polinomial of degree 3 has three roots or zeros.

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

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

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.

Multifunctions can have more than one branch point. In this page we can see a two-valued multifunction with two branch points.

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.

In some cases, a Binomial distribution can be approximated by a Normal distribution with the same mean and variance.

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.

Student's t-distributions were studied by William Gosset(1876-1937) when working with small samples.

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.

In his book 'On Conoids and Spheroids', Archimedes calculated the area of an ellipse. We can see an intuitive approach to Archimedes' ideas.

In his book 'On Conoids and Spheroids', Archimedes calculated the area of an ellipse. It si a good example of a rigorous proof using a double reductio ad absurdum.

Leonardo da Vinci made several drawings of polyhedra for Luca Pacioli's book 'De divina proportione'. Here we can see an adaptation of the dodecahedron.

Leonardo da Vinci made several drawings of polyhedra for Luca Pacioli's book 'De divina proportione'. Here we can see an adaptation of the truncated octahedron.

Leonardo da Vinci made several drawings of polyhedra for Luca Pacioli's book 'De divina proportione'. Here we can see an adaptation of the cuboctahedron.

Leonardo da Vinci made several drawings of polyhedra for Luca Pacioli's book 'De divina proportione'. Here we can see an adaptation of the stellated octahedron (stella octangula).

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.