matematicas visuales visual math
Site Map

Morley Theorem | matematicasVisuales John Conway's proof of Morley's Theorem | matematicasVisuales Wallace-Simson lines | matematicasVisuales Wallace-Simson lines | Demonstration | matematicasVisuales Steiner deltoid | matematicasVisuales Steiner deltoid is a 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 Drawing fifteen degrees angles | matematicasVisuales Dilative rotation | matematicasVisuales
Durer and transformations | matematicasVisuales Equiangular spiral | matematicasVisuales Dilation and rotation in an equiangular spiral | matematicasVisuales Equiangular spiral through two points | matematicasVisuales The Diagonal of a Regular Pentagon and the Golden Ratio | matematicasVisuales Drawing a regular pentagon with ruler and compass | 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
Ellipsograph or Trammel of Archimedes | matematicasVisuales Ellipsograph or Trammel of Archimedes (2) | matematicasVisuales The Astroid as envelope of segments and ellipses | matematicasVisuales The Astroid is a hypocyclioid | 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 tetrahedron | matematicasVisuales Regular dodecahedron | matematicasVisuales Volume of a regular dodecahedron | 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
Truncated tetrahedron | matematicasVisuales Truncations of the cube and octahedron | matematicasVisuales Chamfered Cube | matematicasVisuales The Dodecahedron and the Cube | matematicasVisuales Pyritohedron | matematicasVisuales Rhombic Dodecahedron (1): honeycombs | matematicasVisuales Rhombic Dodecahedron (2): honeycomb minima property | matematicasVisuales
Rhombic Dodecahedron (3): Pyramidated cube | matematicasVisuales Rhombic Dodecahedron (4): Rhombic Dodecahedron made of a cube and six sixth of a cube | 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 Plane developments of geometric bodies: Dodecahedron | matematicasVisuales Plane developments of geometric bodies: Octahedron | matematicasVisuales Plane developments of geometric bodies: Tetrahedron | matematicasVisuales Resources: Building Polyhedra with cardboard (Plane Nets) | matematicasVisuales
Resources: Building polyhedra gluing faces  | matematicasVisuales Resources: How to build polyhedra using paper and rubber bands | matematicasVisuales Resources: Building polyhedra gluing discs  | matematicasVisuales Resources: Acona Biconbi, designed by Bruno Munari  | matematicasVisuales Resources: The golden rectangle and the icosahedron | matematicasVisuales Resources: Modular Origami | matematicasVisuales Resources: Building polyhedra using tubes | matematicasVisuales
Resources: Building polyhedra using Zome | matematicasVisuales Resources: Tensegrity | matematicasVisuales Construcción de poliedros. Cuboctaedro y dodecaedro rómbico: Taller de Talento Matemático de Zaragoza 2014 (Spanish) | matematicasVisuales Cube, octahedron, tetrahedron and other polyhedra: Taller de Talento Matemático Zaragoza,Spain, 2014-2015 (Spanish) | matematicasVisuales Duality: cube and octahedron. Taller de Talento Matemático de Zaragoza, Spain. 2015-2016 XII edition (Spanish) | matematicasVisuales The Cuboctahedron and the truncated octahedron. Taller de Talento Matemático de Zaragoza, Spain. 2016-2017 XIII edition (Spanish) | matematicasVisuales Resources 3d Printing: Tetrahedron | matematicasVisuales
Resources 3d Printing: Cube and Octahedron | matematicasVisuales Geometric sequence | matematicasVisuales Sum of a geometric series of ratio 1/4 | matematicasVisuales Sum of a geometric series of ratio 1/2 | matematicasVisuales Convergence of Series: Integral test | 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 Rational Functions (1): Linear rational functions | matematicasVisuales Rational Functions (2): degree 2 denominator | matematicasVisuales Rational Functions (3): Oblique Asymptote | matematicasVisuales
Rational Functions (4): Asymptotic behavior | 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 Piecewise Linear Functions. Only one piece | matematicasVisuales Piecewise Constant Functions | matematicasVisuales Continuous Piecewise Linear Functions | matematicasVisuales Non continuous Piecewise Linear Functions | matematicasVisuales
Exponentials and Logarithms (1): Exponential Functions | matematicasVisuales Exponentials and Logarithms (2): Logarithm definition as an integral | matematicasVisuales Exponentials and Logarithms (3): One property of the integral of the rectangular hyperbola | matematicasVisuales Exponentials and Logarithms (4): the logarithm of a product | matematicasVisuales Exponentials and Logarithms (5): Approximation of number e | matematicasVisuales Exponentials and Logarithms (6): Two definitions of number e | matematicasVisuales Exponentials and Logarithms (7): The exponential as the inverse of the logarithm | matematicasVisuales
Exponentials and Logarithms (8): Hyperbolas, logarithms and exponencials | matematicasVisuales Mercator and Euler: Logarithm Function | matematicasVisuales Exponentials and Logarithms (9): Radioactive decay (Spanish) | 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 Multiplying two complex numbers | matematicasVisuales The product as a complex plane transformation | matematicasVisuales Complex 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 Leonardo da Vinci: Drawing of a truncated tetrahedron made to Luca Pacioli's De divina proportione. | matematicasVisuales
Leonardo da Vinci:Drawing of an octahedron made to Luca Pacioli's De divina proportione. | matematicasVisuales Leonardo da Vinci:Drawing of a rhombicuboctahedron made to Luca Pacioli's De divina proportione. | matematicasVisuales Durer's approximation of a Regular Pentagon | matematicasVisuales Durer and transformations | matematicasVisuales Kepler: The Area of a Circle | 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 Theorem | matematicasVisuales
The three points of intersection of the adjacent trisectors of the angles of any triangle are the vertices of an equilateral triangle (Morley's triangle)
John Conway's proof of Morley's Theorem | matematicasVisuales
Interactive animation about John Conway's beautiful proof of Morley's Theorem
Wallace-Simson lines | matematicasVisuales
Each point in the circle circunscribed to a triangle give us a line (Wallace-Simson line)
Wallace-Simson lines | Demonstration | matematicasVisuales
Interactive demonstration of the Wallace-Simson line.
Steiner deltoid | matematicasVisuales
The Simson-Wallace lines of a triangle envelops a curve called the Steiner Deltoid.
Steiner deltoid is a 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.
Drawing fifteen degrees angles | matematicasVisuales
Using a ruler and a compass we can draw fifteen degrees angles. These are basic examples of the central and inscribed in a circle angles property.

Plane Transformations
Dilative rotation | matematicasVisuales
A Dilative Rotation is a combination of a rotation an a dilatation from the same point.
Durer and transformations | matematicasVisuales
He studied transformations of images, for example, faces.

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.
Equiangular spiral through two points | matematicasVisuales
There are infinitely many equiangular spirals through two given points.

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'.
Drawing a regular pentagon with ruler and compass | matematicasVisuales
You can draw a regular pentagon given one of its sides constructing the golden ratio with ruler and compass.
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.
Ellipsograph or Trammel of Archimedes | matematicasVisuales
An Ellipsograph is a mechanical device used for drawing ellipses.
Ellipsograph or Trammel of Archimedes (2) | matematicasVisuales
If a straight-line segment is moved in such a way that its extremities travel on two mutually perpendicular straight lines then the midpoint traces out a circle; every other point of the line traces out an ellipse.

More curves
The Astroid as envelope of segments and ellipses | matematicasVisuales
The Astroid is the envelope of a segment of constant length moving with its ends upon two perpendicular lines. It is also the envelope of a family of ellipses, the sum of whose axes is constant.
The Astroid is a hypocyclioid | matematicasVisuales
The Astroid is a particular case of a family of curves called hypocycloids.

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). We can calculate the area of these cross-sections.
Sections in Howard Eves's tetrahedron | matematicasVisuales
In his article 'Two Surprising Theorems on Cavalieri Congruence' Howard Eves describes an interesting tetrahedron. In this page we calculate its cross-section areas and its volume.
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 tetrahedron | matematicasVisuales
Howard Eves's tetrahedron is Cavalieri congruent with a given sphere. You can see that corresponding sections have the same area. Then the volumen of the sphere is the same as the volume of the tetrahedron. And we know how to calculate this volumen.
Regular dodecahedron | matematicasVisuales
Some properties of this platonic solid and how it is related to the golden ratio. Constructing dodecahedra using different techniques.
Volume of a regular dodecahedron | 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.
Truncated tetrahedron | matematicasVisuales
The truncated tetrahedron is an Archimedean solid made by 4 triangles and 4 hexagons.
Truncations of the cube and octahedron | matematicasVisuales
When you truncate a cube you get a truncated cube and a cuboctahedron. If you truncate an octahedron you get a truncated octahedron and a cuboctahedron.
Chamfered Cube | matematicasVisuales
You can chamfer a cube and then you get a polyhedron similar (but not equal) to a truncated octahedron. You can get also a rhombic dodecahedron.
The Dodecahedron and the Cube | matematicasVisuales
A Cube can be inscribed in a Dodecahedron. A Dodecahedron can be seen as a cube with six 'roofs'. You can fold a dodecahedron into a cube.
Pyritohedron | matematicasVisuales
If you fold the six roofs of a regular dodecahedron into a cube there is an empty space. This space can be filled with an irregular dodecahedron composed of identical irregular pentagons (a kind of pyritohedron).

Space Geometry: Rhombic Dodecahedron
Rhombic Dodecahedron (1): honeycombs | matematicasVisuales
Humankind has always been fascinated by how bees build their honeycombs. Kepler related honeycombs with a polyhedron called Rhombic Dodecahedron.
Rhombic Dodecahedron (2): honeycomb minima property | matematicasVisuales
We want to close a hexagonal prism as bees do, using three rhombi. Then, which is the shape of these three rhombi that closes the prism with the minimum surface area?.
Rhombic Dodecahedron (3): Pyramidated cube | matematicasVisuales
Adding six pyramids to a cube you can build new polyhedra with twenty four triangular faces. For specific pyramids you get a Rhombic Dodecahedron that has twelve rhombic faces.
Rhombic Dodecahedron (4): Rhombic Dodecahedron made of a cube and six sixth of a cube | matematicasVisuales
You can build a Rhombic Dodecahedron adding six pyramids to a cube. This fact has several interesting consequences.

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.
Plane developments of geometric bodies: Dodecahedron | matematicasVisuales
The first drawing of a plane net of a regular dodecahedron was published by Dürer in his book 'Underweysung der Messung' ('Four Books of Measurement'), published in 1525 .
Plane developments of geometric bodies: Octahedron | matematicasVisuales
The first drawing of a plane net of a regular octahedron was published by Dürer in his book 'Underweysung der Messung' ('Four Books of Measurement'), published in 1525 .
Plane developments of geometric bodies: Tetrahedron | matematicasVisuales
The first drawing of a plane net of a regular tetrahedron was published by Dürer in his book 'Underweysung der Messung' ('Four Books of Measurement'), published in 1525 .

Resources: Building polyhedra, simple techniques
Resources: Building Polyhedra with cardboard (Plane Nets) | matematicasVisuales
Using cardboard you can draw plane nets and build polyhedra.
Resources: Building polyhedra gluing faces  | matematicasVisuales
Using cardboard you can build beautiful polyhedra cutting polygons and glue them toghether. This is a very simple and effective technique. You can download several templates. Then print, cut and glue: very easy!
Resources: How to build polyhedra using paper and rubber bands | matematicasVisuales
A very simple technique to build complex and colorful polyhedra.
Resources: Building polyhedra gluing discs  | matematicasVisuales
Simple technique to build polyhedra gluing discs made of cardboard or paper.
Resources: Acona Biconbi, designed by Bruno Munari  | matematicasVisuales
Italian designer Bruno Munari conceived 'Acona Biconbi' as a work of sculpture. It is also a beautiful game to play with colors and shapes.
Resources: The golden rectangle and the icosahedron | matematicasVisuales
With three golden rectangles you can build an icosahedron.
Resources: Modular Origami | matematicasVisuales
Modular Origami is a nice technique to build polyhedra.
Resources: Building polyhedra using tubes | matematicasVisuales
Examples of polyhedra built using tubes.
Resources: Building polyhedra using Zome | matematicasVisuales
Examples of polyhedra built using Zome.
Resources: Tensegrity | matematicasVisuales
Examples of polyhedra built using tensegrity.
Construcción de poliedros. Cuboctaedro y dodecaedro rómbico: Taller de Talento Matemático de Zaragoza 2014 (Spanish) | matematicasVisuales
Material for a sesion about polyhedra (Zaragoza, 9th May 2014). Simple techniques to build polyhedra like the tetrahedron, octahedron, the cuboctahedron and the rhombic dodecahedron. We can build a box that is a rhombic dodecahedron.
Cube, octahedron, tetrahedron and other polyhedra: Taller de Talento Matemático Zaragoza,Spain, 2014-2015 (Spanish) | matematicasVisuales
Material for a sesion about polyhedra (Zaragoza, 7th November 2014). We study the octahedron and the tetrahedron and their volumes. The truncated octahedron helps us to this task. We build a cubic box with cardboard and an origami tetrahedron.
Duality: cube and octahedron. Taller de Talento Matemático de Zaragoza, Spain. 2015-2016 XII edition (Spanish) | matematicasVisuales
Material for a sesion about polyhedra (Zaragoza, 23rd Octuber 2015) . Building a cube with cardboard and an origami octahedron.
The Cuboctahedron and the truncated octahedron. Taller de Talento Matemático de Zaragoza, Spain. 2016-2017 XIII edition (Spanish) | matematicasVisuales
Material for a sesion about polyhedra (Zaragoza, 21st October 2016). Instructions to build several geometric bodies.

Resources: Building polyhedra, 3d printing
Resources 3d Printing: Tetrahedron | matematicasVisuales
Building tetraedra using 3d printing. The tetrahedron is a self-dual polyhedron. The center of a tetrahedron.
Resources 3d Printing: Cube and Octahedron | matematicasVisuales
Building cubes and octahedra using 3d printing. Cube and Octahedron are dual polyhedra.

Real Analysis

Sequences and Series
Geometric sequence | matematicasVisuales
Geometric sequences graphic representations. Sum of terms of a geometric sequence and geometric series.
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.
Convergence of Series: Integral test | matematicasVisuales
Using a decreasing positive function you can define series. The integral test is a tool to decide if a series converges o diverges. If a series converges, the integral test provide us lower and upper bounds.
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.

Rational Functions
Rational Functions (1): Linear rational functions | matematicasVisuales
Rational functions can be writen as the quotient of two polynomials. Linear rational functions are the simplest of this kind of functions.
Rational Functions (2): degree 2 denominator | matematicasVisuales
When the denominator of a rational function has degree 2 the function can have two, one or none real singularities.
Rational Functions (3): Oblique Asymptote | matematicasVisuales
For large absolute values of x, some rational functions behave like an oblique straight line, we call this line an oblique or slant asymptote.
Rational Functions (4): Asymptotic behavior | matematicasVisuales
You can add a polynomial to a proper rational function. The end behavior of this rational function is very similar to the 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).

Piecewise Functions
Piecewise Linear Functions. Only one piece | matematicasVisuales
As an introduction to Piecewise Linear Functions we study linear functions restricted to an open interval: their graphs are like segments.
Piecewise Constant Functions | matematicasVisuales
A piecewise function is a function that is defined by several subfunctions. If each piece is a constant function then the piecewise function is called Piecewise constant function or Step function.
Continuous Piecewise Linear Functions | matematicasVisuales
A continuous piecewise linear function is defined by several segments or rays connected, without jumps between them.
Non continuous Piecewise Linear Functions | matematicasVisuales
Graphs of these functions are made of disconnected line segments. There are points where a small change in x produces a sudden jump in the value of the function.

Exponentials and Logarithms
Exponentials and Logarithms (1): Exponential Functions | matematicasVisuales
We can study several properties of exponential functions, their derivatives and an introduction to the number e.
Exponentials and Logarithms (2): Logarithm definition as an integral | matematicasVisuales
Using the integral of the equilateral hyperbola we can define a new function that is the natural logarithm function.
Exponentials and Logarithms (3): One property of the integral of the rectangular hyperbola | matematicasVisuales
The natural logaritm can be defined using the integral of the rectangular hiperbola. In this page we are going to see an important property of this integral. Using this property you can justify that the logarithm of a product is the sum of the logarithms.
Exponentials and Logarithms (4): the logarithm of a product | matematicasVisuales
The main property of a logarithm function is that the logarithm of a product is the sum of the logarithms of the individual factors.
Exponentials and Logarithms (5): Approximation of number e | matematicasVisuales
The logarithm of the number e is equal to 1. Using this definition of the number e we can approximate its value.
Exponentials and Logarithms (6): Two definitions of number e | matematicasVisuales
Constant e is the number whose natural logarithm is 1. It can be defined as a limit of a sequence related with the compound interest. Both definitions for e are equivalent.
Exponentials and Logarithms (7): The exponential as the inverse of the logarithm | matematicasVisuales
After the definition of the natural logarithm function as an integral you can define the exponential function as the inverse function of the logarithm.
Exponentials and Logarithms (8): Hyperbolas, logarithms and exponencials | matematicasVisuales
Different hyperbolas allow us to define different logarithms functions and their inversas, exponentials functions.
Mercator and Euler: Logarithm Function | matematicasVisuales
Mercator published his famous series for the Logarithm Function in 1668. Euler discovered a practical series to calculate.
Exponentials and Logarithms (9): Radioactive decay (Spanish) | matematicasVisuales

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.

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.
Complex 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).
Leonardo da Vinci: Drawing of a truncated tetrahedron 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 tetrahedron.
Leonardo da Vinci:Drawing of an 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 octahedron.
Leonardo da Vinci:Drawing of a rhombicuboctahedron 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 rhombicuboctahedron.

Durer
Durer's approximation of a Regular Pentagon | matematicasVisuales
In his book 'Underweysung der Messung' Durer draw a non-regular pentagon with ruler and a fixed compass. It is a simple construction and a very good approximation of a regular pentagon.
Durer and transformations | matematicasVisuales
He studied transformations of images, for example, faces.

Kepler
Kepler: The Area of a Circle | matematicasVisuales
Kepler used an intuitive infinitesimal approach to calculate the area of a circle.
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.