30th January 2012
Geometry: Plane developments of geometric bodies
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Plane developments of cones and conical frustum. How to calculate the lateral surface area.
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9th January 2012
Geometry: Plane developments of geometric bodies
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Plane net of pyramids cut by an oblique plane.
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2nd December 2011
Geometry: Plane developments of geometric bodies
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Plane net of pyramids and pyramidal frustrum. How to calculate the lateral surface area.
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18th November 2011
Personal: The Game of Life with Nature photos
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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.
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4th November 2011
Geometry: Plane developments of geometric bodies
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We study different cylinders cut by an oblique plane. The section that we get is an ellipse.
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21st October 2011
Geometry: Plane developments of geometric bodies
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We study different cylinders and we can see how they develop into a plane. Then we explain how to calculate the lateral surface area.
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7th October 2011
Geometry: Plane developments of geometric bodies
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Plane nets of prisms with a regular base with different side number cut by an oblique plane.
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30th September 2011
Geometry: Plane developments of geometric bodies
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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.
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15th September 2011
Analysis
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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.
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30th August 2011
Probability
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In some cases, a Binomial distribution can be approximated by a Normal distribution with the same mean and variance.
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3rd August 2011
Probability
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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.
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26th June 2011
Geometry
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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.
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29th May 2011
History: Archimedes and the area of the ellipse
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In his book 'On Conoids and Spheroids', Archimedes calculated the area of an ellipse. We can see an intuitive approach to Archimedes' ideas.
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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.
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29th May 2011
Geometry: Ellipses
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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.
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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.
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29th April 2011
Drawings of Leonardo da Vinci for Luca Pacioli's book 'De divine proportione'
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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.
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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.
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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.
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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).
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29th April 2011
Volume of polyhedra
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A cuboctahedron is an Archimedean solid. It can be seen as made by cutting off the corners of a cube.
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A cuboctahedron is an Archimedean solid. It can be seen as made by cutting off the corners of an octahedron.
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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.
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The stellated octahedron was drawn by Leonardo for Luca Pacioli's book 'De Divina Proportione'. A hundred years later, Kepler named it stella octangula.
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24th February 2011
Geometry
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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.
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21st January 2011
Geometry
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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.
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5th January 2011
Circles
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Central angle in a circle is twice the angle inscribed in the circle.
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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.
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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.
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Interactive 'Mostation' of the property of central and inscribed angles in a circle. The general case is proved.
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18th September 2010
Complex Functions
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Multifunctions can have more than one branch point. In this page we can see a two-valued multifunction with two branch points.
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26th July 2010
Geometry
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The volume of a tetrahedron is one third of the prism that contains it.
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15th July 2010
Complex Functions
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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.
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11th June 2010
Geometry
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The twelve vertices of an icosahedron lie in three golden rectangles. Then we can calculate the volume of an icosahedron
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7th June 2010
Geometry
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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.
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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.
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2nd June 2010
Analysis
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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.
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25th May 2010
Analysis
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One intuitive example of how to sum a geometric series. In this case, we study the geometric series with ratio equal 1/4.
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7th May 2010
Geometry
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These polyhedra pack together to fill space, forming a 3 dimensional space tessellation or tilling.
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28th April 2010
Geometry
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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.
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23th April 2010
Geometry
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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.
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17th March 2010
Complex Functions
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The Complex Cosine Function maps horizontal lines to confocal ellipses.
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28th February 2010
History
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Kepler used an intuitive infinitesimal approach to calculate the area of a circle.
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19th February 2010
Complex Functions
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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.
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5th February 2010
History
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Using Cavalieri's Principle we can calculate the volume of a sphere.
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8th January 2010
History
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Kepler was one mathematician who contributed to the origin of integral calculus. He used infinitesimal techniques for calculating areas and volumes.
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8th December 2010
History
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Studying the volume of a barrel, Kepler solved a problem about maxima in 1615.
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21th November 2009
History
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Mercator published his famous series for the Logarithm Function in 1668. Euler discovered a practical series to calculate.
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16th November 2009
History
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Archimedes show us in 'The Method' how to use the lever law to discover the area of a parabolic segment.
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26th Octuber 2009
History
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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.
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14th October 2009
Complex Functions
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Inversion preserves the magnitud of angles but the sense is reversed. Orthogonal circles are mapped into orthogonal circles
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6th October 2009
Complex Functions
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Inversion is a plane transformation that transform straight lines and circles in straight lines and circles.
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22th September 2009
Complex Functions
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The Complex Exponential Function extends the Real Exponential Function to the complex plane.
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14th September 2009
Personal
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In this new version of The Game of Life invented by John H. Conway we can see more than 100 photos of Nature.
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1st September 2009
Taylor Polynomials
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The complex exponential function is periodic. His power series converges everywhere in the complex plane.
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The power series of the Cosine Function converges everywhere in the complex plane.
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15th June 2009
Taylor Polynomials
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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.
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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.
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We will see how Taylor polynomials approximate the function inside its circle of convergence.
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23th May 2009
Taylor Polynomials
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By increasing the degree, Taylor polynomial approximates the exponential function more and more.
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By increasing the degree, Taylor polynomial approximates the sine function more and more.
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The function is not defined for values less than -1. Taylor polynomials about the origin approximates the function between -1 and 1.
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The function has a singularity at -1. Taylor polynomials about the origin approximates the function between -1 and 1.
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The function has a singularity at -1. Taylor polynomials about the origin approximates the function between -1 and 1.
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8th May 2009
Personal, new section
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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.
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28th February 2009
Space Geometry
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One eighth of a regular dodecahedon of edge 2 has the same volume as a dodecahedron of edge 1.
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26th January 2009
Transformations
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He studied transformations of images, for example, faces.
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In this painting we can see, among lots of interesting things, an anamorphosis of a skull. (In Spanish)
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19th January 2009
Space Geometry
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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.
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10th January 2009
Sequences and Series
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Gamma, the Euler's constant, is defined using a covergent series.
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17th November 2008
Space Geometry
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One eighth of a regular dodecahedon of edge 2 has the same volume as a dodecahedron of edge 1.
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One eighth of a regular dodecahedon of edge 2 has the same volume as a dodecahedron of edge 1.
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8th November 2008
Space Geometry
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The volume of a tetrahedron is one third of the prism that contains it.
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Special sections of a tetrahedron are rectangles (and even squares)
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Howard Eves, mathematician and historian of Mathematics, received the George Polya Award for the article Two Surprising Theorems on Cavalieri Congruence
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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.
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We show a sphere and the Howard Eves's tetrahedron with congruent sections.
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12th August 2008
Random Variables
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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.
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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.
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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.
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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.
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It may be interesting to familiarize ourselves with the probabilities correspondig to different intervals in normal distributions.
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Student's t-distributions were studied by William Gosset(1876-1937) when working with small samples.
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4th August 2007
MatematicasVisuales first English version.