matematicasvisuales visual mathematics home

3rd April 2017

Geometry: Rhombic Dodecahedron
Rhombic Dodecahedron (4): Rhombic Dodecahedron made of a cube and six sixth of a cube | matematicasvisuales |Visual Mathematics
You can build a Rhombic Dodecahedron adding six pyramids to a cube. This fact has several interesting consequences.

6th March 2017

Geometry: Rhombic Dodecahedron
Rhombic Dodecahedron (3): Pyramidated cube | matematicasvisuales |Visual Mathematics
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.

20th February 2017

Geometry: Building polyhedra
Resources 3d Printing: Cube and Octahedron | matematicasvisuales |Visual Mathematics
Building cubes and octahedra using 3d printing. Cube and Octahedron are dual polyhedra.

6th February 2017

Geometry: Rhombic Dodecahedron
Rhombic Dodecahedron (2): honeycomb minima property | matematicasvisuales |Visual Mathematics
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?.

16th January 2017

Geometry: Building polyhedra
Resources 3d Printing: Tetrahedron | matematicasvisuales |Visual Mathematics
Building tetraedra using 3d printing. The tetrahedron is a self-dual polyhedron. The center of a tetrahedron.

9th January 2017

Geometry: Rhombic Dodecahedron
Rhombic Dodecahedron (1): honeycombs | matematicasvisuales |Visual Mathematics
Humankind has always been fascinated by how bees build their honeycombs. Kepler related honeycombs with a polyhedron called Rhombic Dodecahedron.

5th December 2016

Geometry and Art: Bruno Munari
Resources: Acona Biconbi, designed by Bruno Munari  | matematicasvisuales |Visual Mathematics
Italian designer Bruno Munari conceived 'Acona Biconbi' as a work of sculpture. It is also a beautiful game to play with colors and shapes.

21st November 2016

Geometry: Building polyhedra
Resources: Building polyhedra gluing discs  | matematicasvisuales |Visual Mathematics
Simple technique to build polyhedra gluing discs made of cardboard or paper.

7th November 2016

Geometry: Curves
The Astroid is a hypocyclioid | matematicasvisuales |Visual Mathematics
The Astroid is a particular case of a family of curves called hypocycloids.

19th September 2016

Geometry: Polyhedra
The Cuboctahedron and the truncated octahedron. Taller de Talento Matemático de Zaragoza, Spain. 2016-2017 XIII edition (Spanish) | matematicasvisuales |Visual Mathematics
Material for a sesion about polyhedra (Zaragoza, 21st October 2016). Instructions to build several geometric bodies.

5th September 2016

Geometry: Curves
The Astroid as envelope of segments and ellipses | matematicasvisuales |Visual Mathematics
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.

1st August 2016

Geometry: Ellipses
Ellipsograph or Trammel of Archimedes (2) | matematicasvisuales |Visual Mathematics
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.

4th July 2016

Geometry: Ellipses
Ellipsograph or Trammel of Archimedes | matematicasvisuales |Visual Mathematics
An Ellipsograph is a mechanical device used for drawing ellipses.

6th June 2016

Geometry: Pyritohedron
Pyritohedron | matematicasvisuales |Visual Mathematics
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).

2nd May 2016

Geometry: Dodecahedron and cube
The Dodecahedron and the Cube | matematicasvisuales |Visual Mathematics
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.

4th April 2016

History: Leonardo da Vinci
Leonardo da Vinci:Drawing of a rhombicuboctahedron made to Luca Pacioli's De divina proportione. | matematicasvisuales |Visual Mathematics
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.

7th March 2016

Geometry: Plane net of a tetrahedron
Plane developments of geometric bodies: Tetrahedron | matematicasvisuales |Visual Mathematics
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 .

1st February 2016

Geometry: Plane net of an octahedron
Plane developments of geometric bodies: Octahedron | matematicasvisuales |Visual Mathematics
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 .

18th January 2016

History: Leonardo da Vinci
Leonardo da Vinci:Drawing of an octahedron made to Luca Pacioli's De divina proportione. | matematicasvisuales |Visual Mathematics
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.

31st December 2015

Complex Update: Multiplication of Complex Numbers
The product as a complex plane transformation | matematicasvisuales |Visual Mathematics
The multiplication by a complex number is a transformation of the complex plane: dilative rotation.

7th December 2015

Complex Update: Multiplication of Complex Numbers
Multiplying two complex numbers | matematicasvisuales |Visual Mathematics
We can see it as a dilatative rotation.

16th November 2015

Analysis Update: Geometric sequences
Geometric sequence | matematicasvisuales |Visual Mathematics
Geometric sequences graphic representations. Sum of terms of a geometric sequence and geometric series.

2nd November 2015

History: Durer and transformations of faces
Durer and transformations | matematicasvisuales |Visual Mathematics
He studied transformations of images, for example, faces.

28th September 2015

Geometry: Polyhedra
Duality: cube and octahedron. Taller de Talento Matemático de Zaragoza, Spain. 2015-2016 XII edition (Spanish) | matematicasvisuales |Visual Mathematics
Material for a sesion about polyhedra (Zaragoza, 23rd Octuber 2015) . Building a cube with cardboard and an origami octahedron.

21th September 2015

Geometry: Triangles
John Conway's proof of Morley's Theorem | matematicasvisuales |Visual Mathematics
Interactive animation about John Conway's beautiful proof of Morley's Theorem

14th September 2015

Geometry Updates: Triangles
The deltoid and the Morley triangle | matematicasvisuales |Visual Mathematics
Steiner Deltoid and the Morley triangle are related.

7th September 2015

Geometry Updates: Triangles
Morley Theorem | matematicasvisuales |Visual Mathematics
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)

29th June 2015

Geometry Updates: Triangles
Steiner deltoid | matematicasvisuales |Visual Mathematics
The Simson-Wallace lines of a triangle envelops a curve called the Steiner Deltoid.

8th June 2015

Geometry Updates: Triangles
Wallace-Simson lines | Demonstration | matematicasvisuales |Visual Mathematics
Interactive demonstration of the Wallace-Simson line.

6th April 2015

Geometry: Spirals
Equiangular spiral through two points | matematicasvisuales |Visual Mathematics
There are infinitely many equiangular spirals through two given points.

2nd March 2015

Analysis: Piecewise Functions
Non continuous Piecewise Linear Functions | matematicasvisuales |Visual Mathematics
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.

2nd February 2015

Analysis: Piecewise Functions
Continuous Piecewise Linear Functions | matematicasvisuales |Visual Mathematics
A continuous piecewise linear function is defined by several segments or rays connected, without jumps between them.

12th January 2015

Analysis: Piecewise Functions
Piecewise Linear Functions. Only one piece | matematicasvisuales |Visual Mathematics
As an introduction to Piecewise Linear Functions we study linear functions restricted to an open interval: their graphs are like segments.

1st December 2014

Analysis: Piecewise Functions
Piecewise Constant Functions | matematicasvisuales |Visual Mathematics
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.

27th October 2014

Geometry: Polyhedra
Cube, octahedron, tetrahedron and other polyhedra: Taller de Talento Matemático Zaragoza,Spain, 2014-2015 (Spanish) | matematicasvisuales |Visual Mathematics
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.

20th October 2014

Geometry: Polyhedra
Resources: Building polyhedra gluing faces  | matematicasvisuales |Visual Mathematics
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!

6th October 2014

Geometry: Polyhedra
Chamfered Cube | matematicasvisuales |Visual Mathematics
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.

7th July 2014

Geometry: Polyhedra
Truncations of the cube and octahedron | matematicasvisuales |Visual Mathematics
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.

2nd June 2014

Geometry: Polyhedra
Truncated tetrahedron | matematicasvisuales |Visual Mathematics
The truncated tetrahedron is an Archimedean solid made by 4 triangles and 4 hexagons.

27th April 2014

Geometry: Building Polyhedra, cuboctahedron and rhombic dodecahedron (Spanish)
Construcción de poliedros. Cuboctaedro y dodecaedro rómbico: Taller de Talento Matemático de Zaragoza 2014 (Spanish) | matematicasvisuales |Visual Mathematics
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.

7th April 2014

History: Leonardo da Vinci
Leonardo da Vinci: Drawing of a truncated tetrahedron made to Luca Pacioli's De divina proportione. | matematicasvisuales |Visual Mathematics
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.

3rd March 2014

Analysis: Rational Functions
Rational Functions (4): Asymptotic behavior | matematicasvisuales |Visual Mathematics
You can add a polynomial to a proper rational function. The end behavior of this rational function is very similar to the polynomial.

3rd February 2014

Analysis: Rational Functions
Rational Functions (3): Oblique Asymptote | matematicasvisuales |Visual Mathematics
For large absolute values of x, some rational functions behave like an oblique straight line, we call this line an oblique or slant asymptote.

7th January 2014

Analysis: Rational Functions
Rational Functions (2): degree 2 denominator | matematicasvisuales |Visual Mathematics
When the denominator of a rational function has degree 2 the function can have two, one or none real singularities.

2nd December 2013

Analysis: Rational Functions
Rational Functions (1): Linear rational functions | matematicasvisuales |Visual Mathematics
Rational functions can be writen as the quotient of two polynomials. Linear rational functions are the simplest of this kind of functions.

20th October 2013

Analysis: Convergence of Series, the Integral Test
Convergence of Series: Integral test | matematicasvisuales |Visual Mathematics
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.

7th October 2013

History: Durer's construction of a non regular pentagon
Durer's approximation of a Regular Pentagon | matematicasvisuales |Visual Mathematics
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.

16th September 2013

Geometry: Plane net of a dodecahedron
Plane developments of geometric bodies: Dodecahedron | matematicasvisuales |Visual Mathematics
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 .

2nd September 2013

Geometry: Volume of a regular dodecahedron
Volume of a regular dodecahedron | matematicasvisuales |Visual Mathematics
One eighth of a regular dodecahedon of edge 2 has the same volume as a dodecahedron of edge 1.

1st July 2013

Geometry: Drawing a regular pentagon
Drawing a regular pentagon with ruler and compass | matematicasvisuales |Visual Mathematics
You can draw a regular pentagon given one of its sides constructing the golden ratio with ruler and compass.

3rd June 2013

Geometry: Fifteen degrees angle
Drawing fifteen degrees angles | matematicasvisuales |Visual Mathematics
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.

6th May 2013

Geometry: The Golden Ratio
The Diagonal of a Regular Pentagon and the Golden Ratio | matematicasvisuales |Visual Mathematics
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'.

1st April 2013

Analysis: The Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus (2) | matematicasvisuales |Visual Mathematics
The Second Fundamental Theorem of Calculus is a powerful tool for evaluating definite integral (if we know an antiderivative of the function).

4th March 2013

Analysis: The Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus (1) | matematicasvisuales |Visual Mathematics
The Fundamental Theorem of Calculus tell us that every continuous function has an antiderivative and shows how to construct one using the integral.

18th February 2013

Analysis: Powers and Polynomials
Powers with natural exponents (and positive rational exponents) | matematicasvisuales |Visual Mathematics
Power with natural exponents are simple and important functions. Their inverse functions are power with rational exponents (a radical or a nth root)

3th February 2013

Analysis: Integral
Integral of powers with natural exponent | matematicasvisuales |Visual Mathematics
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.

3th January 2013

Analysis: Integral
Monotonic functions are integrable | matematicasvisuales |Visual Mathematics
Monotonic functions in a closed interval are integrable. In these cases we can bound the error we make when approximating the integral using rectangles.

3th December 2012

Analysis: Integral
Indefinite integral | matematicasvisuales |Visual Mathematics
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.

12th November 2012

Analysis: Integral
Definite integral (New version) | matematicasvisuales |Visual Mathematics
The integral concept is associated 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.

22th October 2012

Analysis: Polynomial functions and derivative
Polynomial functions and derivative (5): Antidifferentiation | matematicasvisuales |Visual Mathematics
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.

1st October 2012

Analysis: Powers and polynomials
Polynomial Functions (4): Lagrange interpolating polynomials (New Version) | matematicasvisuales |Visual Mathematics
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.

17th September 2012

Analysis: Polynomial functions and integral
Polynomial functions and integral (3): Lagrange polynomials (General polynomial functions) | matematicasvisuales |Visual Mathematics
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.

27th August 2012

Analysis: Polynomial functions and integral
Polynomial functions and integral (2): Quadratic functions | matematicasvisuales |Visual Mathematics
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.

6th August 2012

Analysis: Polynomial functions and integral
Polynomial functions and integral (1): Linear functions | matematicasvisuales |Visual Mathematics
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.

18th June 2012

Analysis: Polynomial functions and derivative
Polynomial functions and derivative (4): Lagrange polynomials (General polynomial functions) | matematicasvisuales |Visual Mathematics
Lagrange polynomials are polynomials that pases through n given points. We use Lagrange polynomials to explore a general polynomial function and its derivative.

28th May 2012

Analysis: Polynomial functions and derivative
Polynomial functions and derivative (3): Cubic functions | matematicasvisuales |Visual Mathematics
The derivative of a cubic function is a quadratic function, a parabola.

7th May 2012

Analysis: Polynomial functions and derivative
Polynomial functions and derivative (2): Quadratic functions | matematicasvisuales |Visual Mathematics
The derivative of a quadratic function is a linear function, it is to say, a straight line.

16th April 2012

Analysis: Polynomial functions and derivative
Polynomial functions and derivative (1): Linear functions | matematicasvisuales |Visual Mathematics
The derivative of a lineal function is a constant function.

20th March 2012

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

20th February 2012

Geometry: Plane developments of geometric bodies
Plane developments of geometric bodies (8): Cones cut by an oblique plane | matematicasvisuales |Visual Mathematics
Plane developments of cones cut by an oblique plane. The section is an ellipse.

30th January 2012

Geometry: Plane developments of geometric bodies
Plane developments of geometric bodies (7): Cone and conical frustrum | matematicasvisuales |Visual Mathematics
Plane developments of cones and conical frustum. How to calculate the lateral surface area.

9th January 2012

Geometry: Plane developments of geometric bodies
Plane developments of geometric bodies (6): Pyramids cut by an oblique plane | matematicasvisuales |Visual Mathematics
Plane net of pyramids cut by an oblique plane.

2nd December 2011

Geometry: Plane developments of geometric bodies
Plane developments of geometric bodies (5): Pyramid and pyramidal frustrum | matematicasvisuales |Visual Mathematics
Plane net of pyramids and pyramidal frustrum. How to calculate the lateral surface area.

18th November 2011

Personal: The Game of Life with Nature photos
Life Vida (2011 New version, more photos) | matematicasvisuales |Visual Mathematics
In this new version of The Game of Life invented by John H. Conway we can see more than 100 new photos of Nature. Each time you run the application, 36 photos randomly choosen are shown.

4th November 2011

Geometry: Plane developments of geometric bodies
Plane developments of geometric bodies (4): Cylinders cut by an oblique plane | matematicasvisuales |Visual Mathematics
We study different cylinders cut by an oblique plane. The section that we get is an ellipse.

21st October 2011

Geometry: Plane developments of geometric bodies
Plane developments of geometric bodies (3): Cylinders | matematicasvisuales |Visual Mathematics
We study different cylinders and we can see how they develop into a plane. Then we explain how to calculate the lateral surface area.

7th October 2011

Geometry: Plane developments of geometric bodies
Plane developments of geometric bodies (2): Prisms cut by an oblique plane | matematicasvisuales |Visual Mathematics
Plane nets of prisms with a regular base with different side number cut by an oblique plane.

30th September 2011

Geometry: Plane developments of geometric bodies
Plane developments of geometric bodies (1): Nets of prisms | matematicasvisuales |Visual Mathematics
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.

15th September 2011

Analysis
Taylor polynomials (new version) | matematicasvisuales |Visual Mathematics
New version of several pages about Taylor Polynomial with improved mathlets and more images. We start studying several real functions but we need to go to the complex plane to get a better understandig of the concept.

30th August 2011

Probability
Normal approximation to Binomial distribution | matematicasvisuales |Visual Mathematics
In some cases, a Binomial distribution can be approximated by a Normal distribution with the same mean and variance.

3rd August 2011

Probability
Binomial distribution (New Version) | matematicasvisuales |Visual Mathematics
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.

26th June 2011

Geometry
Sections in the sphere (New version) | matematicasvisuales |Visual Mathematics
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 use this result in some applications of Cavalieri's Theorem.

29th May 2011

History: Archimedes and the area of the ellipse
Archimedes and the area of an ellipse: an intuitive approach | matematicasvisuales |Visual Mathematics
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 |Visual Mathematics
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.

29th May 2011

Geometry: Ellipses
Equation of an ellipse | matematicasvisuales |Visual Mathematics
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 |Visual Mathematics
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.

29th April 2011

Drawings of Leonardo da Vinci for Luca Pacioli's book 'De divine proportione'
Leonardo da Vinci: Drawing of a dodecahedron made to Luca Pacioli's De divina proportione. | matematicasvisuales |Visual Mathematics
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 |Visual Mathematics
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 |Visual Mathematics
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 |Visual Mathematics
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).

29th April 2011

Volume of polyhedra
The volume of a cuboctahedron | matematicasvisuales |Visual Mathematics
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 |Visual Mathematics
A cuboctahedron is an Archimedean solid. It can be seen as made by cutting off the corners of an octahedron.
Stellated cuboctahedron | matematicasvisuales |Visual Mathematics
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 |Visual Mathematics
The stellated octahedron was drawn by Leonardo for Luca Pacioli's book 'De Divina Proportione'. A hundred years later, Kepler named it stella octangula.

24th February 2011

Geometry
Standar Paper Size DIN A | matematicasvisuales |Visual Mathematics
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.

21st January 2011

Geometry
The golden ratio | matematicasvisuales |Visual Mathematics
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.

5th January 2011

Circles
Central and inscribed angles in a circle | matematicasvisuales |Visual Mathematics
Central angle in a circle is twice the angle inscribed in the circle.
Central and inscribed angles in a circle | Mostration | Case I | matematicasvisuales |Visual Mathematics
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 |Visual Mathematics
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 |Visual Mathematics
Interactive 'Mostation' of the property of central and inscribed angles in a circle. The general case is proved.

18th September 2010

Complex Functions
Multifunctions: Two branch points | matematicasvisuales |Visual Mathematics
Multifunctions can have more than one branch point. In this page we can see a two-valued multifunction with two branch points.

26th July 2010

Geometry
The volume of the tetrahedron (new version) | matematicasvisuales |Visual Mathematics
The volume of a tetrahedron is one third of the prism that contains it.

15th July 2010

Complex Functions
Multifunctions: Powers with fractional exponent | matematicasvisuales |Visual Mathematics
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.

11th June 2010

Geometry
The icosahedron and its volume | matematicasvisuales |Visual Mathematics
The twelve vertices of an icosahedron lie in three golden rectangles. Then we can calculate the volume of an icosahedron

7th June 2010

Geometry
Hexagonal section of a cube | matematicasvisuales |Visual Mathematics
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 |Visual Mathematics
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.

2nd June 2010

Analysis
Sum of a geometric series of ratio 1/2 | matematicasvisuales |Visual Mathematics
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.

25th May 2010

Analysis
Geometric series sum (New Version) | matematicasvisuales |Visual Mathematics
One intuitive example of how to sum a geometric series. In this case, we study the geometric series with ratio equal 1/4.

7th May 2010

Geometry
The truncated octahedron is a space-filling polyhedron | matematicasvisuales |Visual Mathematics
These polyhedra pack together to fill space, forming a 3 dimensional space tessellation or tilling.

28th April 2010

Geometry
Volume of an Octahedron (New Version) | matematicasvisuales |Visual Mathematics
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.

23th April 2010

Geometry
The volume of a truncated octahedron | matematicasvisuales |Visual Mathematics
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.

17th March 2010

Complex Functions
The Complex Cosine Function: mapping an horizontal line | matematicasvisuales |Visual Mathematics
The Complex Cosine Function maps horizontal lines to confocal ellipses.

28th February 2010

History
Kepler: The Area of a Circle | matematicasvisuales |Visual Mathematics
Kepler used an intuitive infinitesimal approach to calculate the area of a circle.

19th February 2010

Complex Functions
The Complex Cosine Function | matematicasvisuales |Visual Mathematics
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.

5th February 2010

History
Cavalieri: The volume of a sphere | matematicasvisuales |Visual Mathematics
Using Cavalieri's Principle we can calculate the volume of a sphere.

8th January 2010

History
Kepler: The volume of a wine barrel | matematicasvisuales |Visual Mathematics
Kepler was one mathematician who contributed to the origin of integral calculus. He used infinitesimal techniques for calculating areas and volumes.

8th December 2010

History
Kepler: The best proportions for a wine barrel | matematicasvisuales |Visual Mathematics
Studying the volume of a barrel, Kepler solved a problem about maxima in 1615.

21th November 2009

History
Mercator and Euler: Logarithm Function | matematicasvisuales |Visual Mathematics
Mercator published his famous series for the Logarithm Function in 1668. Euler discovered a practical series to calculate.

16th November 2009

History
Archimedes' Method to calculate the area of a parabolic segment | matematicasvisuales |Visual Mathematics
Archimedes show us in 'The Method' how to use the lever law to discover the area of a parabolic segment.

26th Octuber 2009

History
Pythagoras' Theorem in a tiling | matematicasvisuales |Visual Mathematics
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.

14th October 2009

Complex Functions
Inversion: an anticonformal transformation | matematicasvisuales |Visual Mathematics
Inversion preserves the magnitud of angles but the sense is reversed. Orthogonal circles are mapped into orthogonal circles

6th October 2009

Complex Functions
Inversion | matematicasvisuales |Visual Mathematics
Inversion is a plane transformation that transform straight lines and circles in straight lines and circles.

22th September 2009

Complex Functions
The Complex Exponential Function | matematicasvisuales |Visual Mathematics
The Complex Exponential Function extends the Real Exponential Function to the complex plane.

14th September 2009

Personal
Life Vida (New version, more photos) | matematicasvisuales |Visual Mathematics
In this new version of The Game of Life invented by John H. Conway we can see more than 100 photos of Nature.

1st September 2009

Taylor Polynomials
Taylor polynomials: Complex Exponential Function | matematicasvisuales |Visual Mathematics
The complex exponential function is periodic. His power series converges everywhere in the complex plane.
Taylor polynomials: Complex Cosine Function | matematicasvisuales |Visual Mathematics
The power series of the Cosine Function converges everywhere in the complex plane.

15th June 2009

Taylor Polynomials
Taylor polynomials (6): Rational function with two real singularities | matematicasvisuales |Visual Mathematics
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 |Visual Mathematics
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.
Taylor polynomials: Rational function with two complex singularities | matematicasvisuales |Visual Mathematics
We will see how Taylor polynomials approximate the function inside its circle of convergence.

23th May 2009

Taylor Polynomials
Taylor polynomials (1): Exponential function | matematicasvisuales |Visual Mathematics
By increasing the degree, Taylor polynomial approximates the exponential function more and more.
Taylor polynomials (2): Sine function | matematicasvisuales |Visual Mathematics
By increasing the degree, Taylor polynomial approximates the sine function more and more.
Taylor polynomials (3): Square root | matematicasvisuales |Visual Mathematics
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 |Visual Mathematics
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 |Visual Mathematics
The function has a singularity at -1. Taylor polynomials about the origin approximates the function between -1 and 1.

8th May 2009

Personal, new section
Life Vida | matematicasvisuales |Visual Mathematics
Te Game of Life was invented by John H. Conway. It is one of the most famous bidimensional cellular automaton. Using a colony we can see some photographs about Nature.

28th February 2009

Space Geometry
Regular dodecahedron | matematicasvisuales |Visual Mathematics
Some properties of this platonic solid and how it is related to the golden ratio. Constructing dodecahedra using different techniques.

26th January 2009

Transformations
Durer and transformations | matematicasvisuales |Visual Mathematics
He studied transformations of images, for example, faces.
Los Embajadores de Holbein el Joven | matematicasvisuales |Visual Mathematics
In this painting we can see, among lots of interesting things, an anamorphosis of a skull. (In Spanish)

19th January 2009

Space Geometry
Volume of an octahedron | matematicasvisuales |Visual Mathematics
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.

10th January 2009

Sequences and Series
Gamma, Euler's constant | matematicasvisuales |Visual Mathematics
Gamma, the Euler's constant, is defined using a covergent series.

17th November 2008

Space Geometry
Regular dodecahedron | matematicasvisuales |Visual Mathematics
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 |Visual Mathematics
One eighth of a regular dodecahedon of edge 2 has the same volume as a dodecahedron of edge 1.

8th November 2008

Space Geometry
The volume of the tetrahedron | matematicasvisuales |Visual Mathematics
The volume of a tetrahedron is one third of the prism that contains it.
Sections on a tetrahedron | matematicasvisuales |Visual Mathematics
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 |Visual Mathematics
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 |Visual Mathematics
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 |Visual Mathematics
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.

12th August 2008

Random Variables
Binomial distribution | matematicasvisuales |Visual Mathematics
When modeling a situation where there are n independent trials with a constant probability p of success in each test we use a binomial distribution.
Poisson distribution | matematicasvisuales |Visual Mathematics
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 |Visual Mathematics
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 |Visual Mathematics
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 |Visual Mathematics
It may be interesting to familiarize ourselves with the probabilities correspondig to different intervals in normal distributions.
Student's t-distributions | matematicasvisuales |Visual Mathematics
Student's t-distributions were studied by William Gosset(1876-1937) when working with small samples.
Calculating probabilities in t Student distributions (Spanish) | matematicasvisuales |Visual Mathematics

4th August 2007

MatematicasVisuales first English version.