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Morley triangle | matematicasVisuales Wallace-Simson lines | matematicasVisuales Wallace-Simson lines | Mostration | matematicasVisuales Steiner deltoid | matematicasVisuales Steiner deltoid is an hypocycloid | matematicasVisuales The deltoid and the Morley triangle | matematicasVisuales Pythagoras' Theorem in a tiling | matematicasVisuales
Central and inscribed angles in a circle | matematicasVisuales Central and inscribed angles in a circle | Mostration | Case I | matematicasVisuales Central and inscribed angles in a circle | Mostration | Case II | matematicasVisuales Central and inscribed angles in a circle | Mostration | General Case | matematicasVisuales Dilative rotation | matematicasVisuales Durer | matematicasVisuales Los Embajadores de Holbein el Joven | matematicasVisuales
Equiangular spiral | matematicasVisuales Dilation and rotation in an equiangular spiral | matematicasVisuales The Diagonal of a Regular Pentagon and the Golden Ratio | matematicasVisuales The golden ratio | matematicasVisuales The golden rectangle | matematicasVisuales The golden rectangle and the dilative rotation | matematicasVisuales The golden rectangle and two equiangular spirals | matematicasVisuales
The golden spiral | matematicasVisuales Standar Paper Size DIN A | matematicasVisuales Equation of an ellipse | matematicasVisuales Ellipse and its foci | matematicasVisuales The volume of the tetrahedron | matematicasVisuales Sections on a tetrahedron | matematicasVisuales Sections in Howard Eves's tetrahedron | matematicasVisuales
Sections in the sphere | matematicasVisuales Surprising Cavalieri congruence between a sphere and a tetrahedronn | matematicasVisuales Regular dodecahedron | matematicasVisuales Volume of a regular dodecahedron (Flash version) | matematicasVisuales Volume of an octahedron | matematicasVisuales The icosahedron and its volume | matematicasVisuales The volume of a truncated octahedron | matematicasVisuales
The truncated octahedron is a space-filling polyhedron | matematicasVisuales Hexagonal section of a cube | matematicasVisuales A truncated octahedron made by eight half cubes | matematicasVisuales The volume of a cuboctahedron | matematicasVisuales The volume of a cuboctahedron (II) | matematicasVisuales Stellated cuboctahedron | matematicasVisuales The volume of an stellated octahedron (stella octangula) | matematicasVisuales
Plane developments of geometric bodies (1): Nets of prisms | matematicasVisuales Plane developments of geometric bodies (2): Prisms cut by an oblique plane | matematicasVisuales Plane developments of geometric bodies (3): Cylinders | matematicasVisuales Plane developments of geometric bodies (4): Cylinders cut by an oblique plane | matematicasVisuales Plane developments of geometric bodies (5): Pyramid and pyramidal frustrum | matematicasVisuales Plane developments of geometric bodies (6): Pyramids cut by an oblique plane | matematicasVisuales Plane developments of geometric bodies (7): Cone and conical frustrum | matematicasVisuales
Plane developments of geometric bodies (8): Cones cut by an oblique plane | matematicasVisuales Building polyhedra. Simple techniques (in Spanish) | matematicasVisuales Geometric sequence | matematicasVisuales Sum of a geometric series of ratio 1/4 | matematicasVisuales Sum of a geometric series of ratio 1/2 | matematicasVisuales Gamma, Euler's constant | matematicasVisuales Polynomial Functions (1): Linear functions | matematicasVisuales
Powers with natural exponents (and positive rational exponents) | matematicasVisuales Polynomial Functions (2): Quadratic functions | matematicasVisuales Polynomial Functions (3): Cubic functions | matematicasVisuales Polynomial Functions (4): Lagrange interpolating polynomial | matematicasVisuales Polynomial functions and derivative (1): Linear functions | matematicasVisuales Polynomial functions and derivative (2): Quadratic functions | matematicasVisuales Polynomial functions and derivative (3): Cubic functions | matematicasVisuales
Polynomial functions and derivative (4): Lagrange polynomials (General polynomial functions) | matematicasVisuales Polynomial functions and derivative (5): Antidifferentiation | matematicasVisuales Definite integral | matematicasVisuales Indefinite integral | matematicasVisuales Monotonic functions are integrable | matematicasVisuales Integral of powers with natural exponent | matematicasVisuales Archimedes' Method to calculate the area of a parabolic segment | matematicasVisuales
Kepler: The best proportions for a wine barrel | matematicasVisuales Polynomial functions and integral (1): Linear functions | matematicasVisuales Polynomial functions and integral (2): Quadratic functions | matematicasVisuales Polynomial functions and integral (3): Lagrange polynomials (General polynomial functions) | matematicasVisuales The Fundamental Theorem of Calculus (1) | matematicasVisuales The Fundamental Theorem of Calculus (2) | matematicasVisuales Taylor polynomials (1): Exponential function | matematicasVisuales
Mercator and Euler: Logarithm Function | matematicasVisuales Taylor polynomials (2): Sine function | matematicasVisuales Taylor polynomials (3): Square root | matematicasVisuales Taylor polynomials (4): Rational function 1 | matematicasVisuales Taylor polynomials (5): Rational function 2 | matematicasVisuales Taylor polynomials (6): Rational function with two real singularities | matematicasVisuales Taylor polynomials (7): Rational function without real singularities | matematicasVisuales
Una propiedad de la integral de la hipérbola (Spanish) | matematicasVisuales Una propiedad de la integral de la hipérbola | 'Mostración'(Spanish) | matematicasVisuales El logaritmo de un producto (Spanish) | matematicasVisuales Definición de logaritmo como una integral (Spanish) | matematicasVisuales Mercator and Euler: Logarithm Function | matematicasVisuales Approximation of number e (Spanish) | matematicasVisuales Two definitions of number e (Spanish) | matematicasVisuales
The exponential as the inverse of the logarithm (Spanish) | matematicasVisuales Hipérbolas, logaritmos y exponenciales (Spanish) | matematicasVisuales Exponencial functions (Spanish) | matematicasVisuales Radioactiv desintegration (Spanish) | matematicasVisuales Multiplying two complex numbers | matematicasVisuales The product as a complex plane transformation | matematicasVisuales Geometric sequence | matematicasVisuales
Complex Polynomial Functions(1): Powers with natural exponent | matematicasVisuales Complex Polynomial Functions(2): Polynomial of degree 2 | matematicasVisuales Complex Polynomial Functions(3): Polynomial of degree 3 | matematicasVisuales Complex Polynomial Functions(4): Polynomial of degree n | matematicasVisuales Complex Polynomial Functions(5): Polynomial of degree n (variant) | matematicasVisuales Cero and polo (Spanish) | matematicasVisuales Cero and polo (variant) (Spanish) | matematicasVisuales
Moebius transformations (Spanish) | matematicasVisuales The Complex Exponential Function | matematicasVisuales The Complex Cosine Function | matematicasVisuales The Complex Cosine Function: mapping an horizontal line | matematicasVisuales Inversion | matematicasVisuales Inversion: an anticonformal transformation | matematicasVisuales Multifunctions: Powers with fractional exponent | matematicasVisuales
Multifunctions: Two branch points | matematicasVisuales Taylor polynomials: Rational function with two complex singularities | matematicasVisuales Taylor polynomials: Complex Exponential Function | matematicasVisuales Taylor polynomials: Complex Cosine Function | matematicasVisuales Binomial distribution | matematicasVisuales Normal approximation to Binomial distribution | matematicasVisuales Poisson distribution | matematicasVisuales
Normal distribution | matematicasVisuales One, two and three standar deviations | matematicasVisuales Calculating probabilities in Normal distributions | matematicasVisuales Student's t-distributions | matematicasVisuales Calculating probabilities in t Student distributions (Spanish) | matematicasVisuales Pythagoras' Theorem in a tiling | matematicasVisuales Archimedes' Method to calculate the area of a parabolic segment | matematicasVisuales
Archimedes and the area of an ellipse: an intuitive approach | matematicasVisuales Archimedes and the area of an ellipse: Demonstration | matematicasVisuales Leonardo da Vinci: Drawing of a dodecahedron made to Luca Pacioli's De divina proportione. | matematicasVisuales Leonardo da Vinci: Drawing of a truncated octahedron made to Luca Pacioli's De divina proportione. | matematicasVisuales Leonardo da Vinci: Drawing of a cuboctahedron made to Luca Pacioli's De divina proportione. | matematicasVisuales Leonardo da Vinci:Drawing of an stellated octahedron (stella octangula) made to Luca Pacioli's De divina proportione. | matematicasVisuales Kepler: The Area of a Circle | matematicasVisuales
Kepler: Surface and volume of a sphere | matematicasVisuales Kepler: The volume of a wine barrel | matematicasVisuales Kepler: The best proportions for a wine barrel | matematicasVisuales Cavalieri: The volume of a sphere | matematicasVisuales Mercator and Euler: Logarithm Function | matematicasVisuales

Geometry

Triangles
Morley triangle | matematicasVisuales
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 | matematicasVisuales
Each point in the circle circunscribed to a triangle give us a line (Wallace-Simson line)
Wallace-Simson lines | Mostration | matematicasVisuales
Interactive 'Mostration' of the Wallace-Simson line.
Steiner deltoid | matematicasVisuales
All the Wallace-Simson lines form a deltoid (Steiner Deltoid).
Steiner deltoid is an hypocycloid | matematicasVisuales
Steiner deltoid is a hypocycloid related with the nine point circle of a triangle.
The deltoid and the Morley triangle | matematicasVisuales
Steiner Deltoid and the Morley triangle are related.
Pythagoras' Theorem in a tiling | matematicasVisuales
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 | matematicasVisuales
Central angle in a circle is twice the angle inscribed in the circle.
Central and inscribed angles in a circle | Mostration | Case I | matematicasVisuales
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.
Central and inscribed angles in a circle | Mostration | Case II | matematicasVisuales
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.
Central and inscribed angles in a circle | Mostration | General Case | matematicasVisuales
Interactive 'Mostation' of the property of central and inscribed angles in a circle. The general case is proved.

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

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

The Golden Ratio
The Diagonal of a Regular Pentagon and the Golden Ratio | matematicasVisuales
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'.
The golden ratio | matematicasVisuales
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.
The golden rectangle | matematicasVisuales
A golden rectangle is made of an square and another golden rectangle.
The golden rectangle and the dilative rotation | matematicasVisuales
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 | matematicasVisuales
Two equiangular spirals contains all vertices of golden rectangles.
The golden spiral | matematicasVisuales
The golden spiral is a good approximation of an equiangular spiral.

Proportions
Standar Paper Size DIN A | matematicasVisuales
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.

Ellipses
Equation of an ellipse | matematicasVisuales
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.
Ellipse and its foci | matematicasVisuales
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.

Space Geometry
The volume of the tetrahedron | matematicasVisuales
The volume of a tetrahedron is one third of the prism that contains it.
Sections on a tetrahedron | matematicasVisuales
Special sections of a tetrahedron are rectangles (and even squares)
Sections in Howard Eves's tetrahedron | matematicasVisuales
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 | matematicasVisuales
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.
Surprising Cavalieri congruence between a sphere and a tetrahedronn | matematicasVisuales
We show a sphere and the Howard Eves's tetrahedron with congruent sections.
Regular dodecahedron | matematicasVisuales
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) | matematicasVisuales
One eighth of a regular dodecahedon of edge 2 has the same volume as a dodecahedron of edge 1.
Volume of an octahedron | matematicasVisuales
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 | matematicasVisuales
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 | matematicasVisuales
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 | matematicasVisuales
These polyhedra pack together to fill space, forming a 3 dimensional space tessellation or tilling.
Hexagonal section of a cube | matematicasVisuales
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 | matematicasVisuales
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.
The volume of a cuboctahedron | matematicasVisuales
A cuboctahedron is an Archimedean solid. It can be seen as made by cutting off the corners of a cube.
The volume of a cuboctahedron (II) | matematicasVisuales
A cuboctahedron is an Archimedean solid. It can be seen as made by cutting off the corners of an octahedron.
Stellated cuboctahedron | matematicasVisuales
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 volume of an stellated octahedron (stella octangula) | matematicasVisuales
The stellated octahedron was drawn by Leonardo for Luca Pacioli's book 'De Divina Proportione'. A hundred years later, Kepler named it stella octangula.

Plane developments of geometric bodies
Plane developments of geometric bodies (1): Nets of prisms | matematicasVisuales
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 developments of geometric bodies (2): Prisms cut by an oblique plane | matematicasVisuales
Plane nets of prisms with a regular base with different side number cut by an oblique plane.
Plane developments of geometric bodies (3): Cylinders | matematicasVisuales
We study different cylinders and we can see how they develop into a plane. Then we explain how to calculate the lateral surface area.
Plane developments of geometric bodies (4): Cylinders cut by an oblique plane | matematicasVisuales
We study different cylinders cut by an oblique plane. The section that we get is an ellipse.
Plane developments of geometric bodies (5): Pyramid and pyramidal frustrum | matematicasVisuales
Plane net of pyramids and pyramidal frustrum. How to calculate the lateral surface area.
Plane developments of geometric bodies (6): Pyramids cut by an oblique plane | matematicasVisuales
Plane net of pyramids cut by an oblique plane.
Plane developments of geometric bodies (7): Cone and conical frustrum | matematicasVisuales
Plane developments of cones and conical frustum. How to calculate the lateral surface area.
Plane developments of geometric bodies (8): Cones cut by an oblique plane | matematicasVisuales
Plane developments of cones cut by an oblique plane. The section is an ellipse.

Building polyhedra. Simple techniques
Building polyhedra. Simple techniques (in Spanish) | matematicasVisuales
Several pages about simple techniques for building polyhedra: cardboard, origami, tubes, zome, tensegrity.

Real Analysis

Sequences and Series
Geometric sequence | matematicasVisuales
Geometric sequences graphic representations
Sum of a geometric series of ratio 1/4 | matematicasVisuales
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 | matematicasVisuales
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 | matematicasVisuales
Gamma, the Euler's constant, is defined using a covergent series.

Powers and Polynomials
Polynomial Functions (1): Linear functions | matematicasVisuales
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.
Powers with natural exponents (and positive rational exponents) | matematicasVisuales
Power with natural exponents are simple and important functions. Their inverse functions are power with rational exponents (a radical or a nth root)
Polynomial Functions (2): Quadratic functions | matematicasVisuales
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.
Polynomial Functions (3): Cubic functions | matematicasVisuales
Polynomials of degree 3 are cubic functions. A real cubic function always crosses the x-axis at least once.
Polynomial Functions (4): Lagrange interpolating polynomial | matematicasVisuales
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.

Polynomial functions and derivative
Polynomial functions and derivative (1): Linear functions | matematicasVisuales
The derivative of a lineal function is a constant function.
Polynomial functions and derivative (2): Quadratic functions | matematicasVisuales
The derivative of a quadratic function is a linear function, it is to say, a straight line.
Polynomial functions and derivative (3): Cubic functions | matematicasVisuales
The derivative of a cubic function is a quadratic function, a parabola.
Polynomial functions and derivative (4): Lagrange polynomials (General polynomial functions) | matematicasVisuales
Lagrange polynomials are polynomials that pases through n given points. We use Lagrange polynomials to explore a general polynomial function and its derivative.
Polynomial functions and derivative (5): Antidifferentiation | matematicasVisuales
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.

Integral
Definite integral | matematicasVisuales
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.
Indefinite integral | matematicasVisuales
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 are integrable | matematicasVisuales
Monotonic functions in a closed interval are integrable. In these cases we can bound the error we make when approximating the integral using rectangles.
Integral of powers with natural exponent | matematicasVisuales
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' Method to calculate the area of a parabolic segment | matematicasVisuales
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 | matematicasVisuales
Studying the volume of a barrel, Kepler solved a problem about maxima in 1615.

Polynomial functions and integral
Polynomial functions and integral (1): Linear functions | matematicasVisuales
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.
Polynomial functions and integral (2): Quadratic functions | matematicasVisuales
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.
Polynomial functions and integral (3): Lagrange polynomials (General polynomial functions) | matematicasVisuales
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
The Fundamental Theorem of Calculus (1) | matematicasVisuales
The Fundamental Theorem of Calculus tell us that every continuous function has an antiderivative and shows how to construct one using the integral.
The Fundamental Theorem of Calculus (2) | matematicasVisuales
The Second Fundamental Theorem of Calculus is a powerful tool for evaluating definite integral (if we know an antiderivative of the function).

Taylor's Polynomials
Taylor polynomials (1): Exponential function | matematicasVisuales
By increasing the degree, Taylor polynomial approximates the exponential function more and more.
Mercator and Euler: Logarithm Function | matematicasVisuales
Mercator published his famous series for the Logarithm Function in 1668. Euler discovered a practical series to calculate.
Taylor polynomials (2): Sine function | matematicasVisuales
By increasing the degree, Taylor polynomial approximates the sine function more and more.
Taylor polynomials (3): Square root | matematicasVisuales
The function is not defined for values less than -1. Taylor polynomials about the origin approximates the function between -1 and 1.
Taylor polynomials (4): Rational function 1 | matematicasVisuales
The function has a singularity at -1. Taylor polynomials about the origin approximates the function between -1 and 1.
Taylor polynomials (5): Rational function 2 | matematicasVisuales
The function has a singularity at -1. Taylor polynomials about the origin approximates the function between -1 and 1.
Taylor polynomials (6): Rational function with two real singularities | matematicasVisuales
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 (7): Rational function without real singularities | matematicasVisuales
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) | matematicasVisuales
Una propiedad de la integral de la hipérbola | 'Mostración'(Spanish) | matematicasVisuales
El logaritmo de un producto (Spanish) | matematicasVisuales
Definición de logaritmo como una integral (Spanish) | matematicasVisuales
Mercator and Euler: Logarithm Function | matematicasVisuales
Mercator published his famous series for the Logarithm Function in 1668. Euler discovered a practical series to calculate.
Approximation of number e (Spanish) | matematicasVisuales
Two definitions of number e (Spanish) | matematicasVisuales
The exponential as the inverse of the logarithm (Spanish) | matematicasVisuales
Hipérbolas, logaritmos y exponenciales (Spanish) | matematicasVisuales
Exponencial functions (Spanish) | matematicasVisuales
Radioactiv desintegration (Spanish) | matematicasVisuales

Complex Analysis

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

Complex Functions
Complex Polynomial Functions(1): Powers with natural exponent | matematicasVisuales
Complex power functions with natural exponent have a zero (or root) of multiplicity n in the origin.
Complex Polynomial Functions(2): Polynomial of degree 2 | matematicasVisuales
A polynomial of degree 2 has two zeros or roots. In this representation you can see Cassini ovals and a lemniscate.
Complex Polynomial Functions(3): Polynomial of degree 3 | matematicasVisuales
A complex polinomial of degree 3 has three roots or zeros.
Complex Polynomial Functions(4): Polynomial of degree n | matematicasVisuales
Every complex polynomial of degree n has n zeros or roots.
Complex Polynomial Functions(5): Polynomial of degree n (variant) | matematicasVisuales
Every complex polynomial of degree n has n zeros or roots.
Cero and polo (Spanish) | matematicasVisuales
Podemos modificar las multiplicidades del cero y del polo de estas funciones sencillas.
Cero and polo (variant) (Spanish) | matematicasVisuales
Moebius transformations (Spanish) | matematicasVisuales
Una primera aproximación a estas transformaciones. Representación de dos haces coaxiales de circunferencias ortogonales.
The Complex Exponential Function | matematicasVisuales
The Complex Exponential Function extends the Real Exponential Function to the complex plane.
The Complex Cosine Function | matematicasVisuales
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 | matematicasVisuales
The Complex Cosine Function maps horizontal lines to confocal ellipses.
Inversion | matematicasVisuales
Inversion is a plane transformation that transform straight lines and circles in straight lines and circles.
Inversion: an anticonformal transformation | matematicasVisuales
Inversion preserves the magnitud of angles but the sense is reversed. Orthogonal circles are mapped into orthogonal circles
Multifunctions: Powers with fractional exponent | matematicasVisuales
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: Two branch points | matematicasVisuales
Multifunctions can have more than one branch point. In this page we can see a two-valued multifunction with two branch points.

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

Probability

Random Variables
Binomial distribution | matematicasVisuales
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.
Normal approximation to Binomial distribution | matematicasVisuales
In some cases, a Binomial distribution can be approximated by a Normal distribution with the same mean and variance.
Poisson distribution | matematicasVisuales
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 | matematicasVisuales
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 | matematicasVisuales
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 | matematicasVisuales
It may be interesting to familiarize ourselves with the probabilities correspondig to different intervals in normal distributions.
Student's t-distributions | matematicasVisuales
Student's t-distributions were studied by William Gosset(1876-1937) when working with small samples.
Calculating probabilities in t Student distributions (Spanish) | matematicasVisuales

History

Pitagoras
Pythagoras' Theorem in a tiling | matematicasVisuales
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 | matematicasVisuales
Archimedes show us in 'The Method' how to use the lever law to discover the area of a parabolic segment.
Archimedes and the area of an ellipse: an intuitive approach | matematicasVisuales
In his book 'On Conoids and Spheroids', Archimedes calculated the area of an ellipse. We can see an intuitive approach to Archimedes' ideas.
Archimedes and the area of an ellipse: Demonstration | matematicasVisuales
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's drawings for Luca Pacioli's book 'De divina proportione'
Leonardo da Vinci: Drawing of a dodecahedron made to Luca Pacioli's De divina proportione. | matematicasVisuales
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: Drawing of a truncated octahedron made to Luca Pacioli's De divina proportione. | matematicasVisuales
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: Drawing of a cuboctahedron made to Luca Pacioli's De divina proportione. | matematicasVisuales
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:Drawing of an stellated octahedron (stella octangula) made to Luca Pacioli's De divina proportione. | matematicasVisuales
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
Kepler: The Area of a Circle | matematicasVisuales
Kepler used an intuitive infinitesimal approach to calculate the area of a circle.
Kepler: Surface and volume of a sphere | matematicasVisuales
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 | matematicasVisuales
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 | matematicasVisuales
Studying the volume of a barrel, Kepler solved a problem about maxima in 1615.

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

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