Geometry

Triangles


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)


Interactive animation about John Conway's beautiful proof of Morley's Theorem


Each point in the circle circunscribed to a triangle give us a line (WallaceSimson line)


Interactive demonstration of the WallaceSimson line.


The SimsonWallace lines of a triangle envelops a curve called the Steiner Deltoid.


Steiner deltoid is a hypocycloid related with the nine point circle of a triangle.


Steiner Deltoid and the Morley triangle are related.


Demonstration of Pythagoras Theorem inspired in Euclid.


Dynamic demonstration of the Pythagorean Theorem by Hermann Baravalle.


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 angle in a circle is twice the angle inscribed in the circle.


Interactive 'Mostation' of the property of central and inscribed angles in a circle. Case I: When the arc is half a circle the inscribed angle is a right angle.


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


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


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.


If a hexagon is inscribed in a circle, the three pairs of opposite sides meet in collinear points.

Plane Transformations


A Dilative Rotation is a combination of a rotation an a dilatation from the same point.


He studied transformations of images, for example, faces.

Spirals


In an equiangular spiral the angle between the position vector and the tangent is constant.


Two transformations of an equiangular spiral with the same general efect.


There are infinitely many equiangular spirals through two given points.

The Golden Ratio


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'.


You can draw a regular pentagon given one of its sides constructing the golden ratio with ruler and compass.


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


A golden rectangle is made of an square and another golden rectangle.


A golden rectangle is made of an square an another golden rectangle. These rectangles are related through an dilative rotation.


Two equiangular spirals contains all vertices of golden rectangles.


The golden spiral is a good approximation of an equiangular spiral.

Proportions


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


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


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


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


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


An Ellipsograph is a mechanical device used for drawing ellipses.


If a straightline 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.


The section of a cylinder by a plane cutting its axis at a single point is an ellipse. A beautiful demonstration uses Dandelin Spheres.


Durer was the first who published in german a method to draw ellipses as cone sections.


Durer made a mistake when he explanined how to draw ellipses. We can prove, using only basic properties, that the ellipse has not an egg shape .

More curves


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 particular case of a family of curves called hypocycloids.

Space Geometry


The volume of a tetrahedron is one third of the prism that contains it.


Special sections of a tetrahedron are rectangles (and even squares). We can calculate the area of these crosssections.


In his article 'Two Surprising Theorems on Cavalieri Congruence' Howard Eves describes an interesting tetrahedron. In this page we calculate its crosssection areas and its volume.


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.


Some properties of this platonic solid and how it is related to the golden ratio. Constructing dodecahedra using different techniques.


One eighth of a regular dodecahedon of edge 2 has the same volume as a dodecahedron of edge 1.


The volume of an octahedron is four times the volume of a tetrahedron. It is easy to calculate and then we can get the volume of a tetrahedron.


The twelve vertices of an icosahedron lie in three golden rectangles. Then we can calculate the volume of an icosahedron


The truncated octahedron is an Archimedean solid. It has 8 regular hexagonal faces and 6 square faces. Its volume can be calculated knowing the volume of an octahedron.


These polyhedra pack together to fill space, forming a 3 dimensional space tessellation or tilling.


We can cut in half a cube by a plane and get a section that is a regular hexagon. Using eight of this pieces we can made a truncated octahedron.


Using eight half cubes we can make a truncated octahedron. The cube tesselate the space an so do the truncated octahedron. We can calculate the volume of a truncated octahedron.


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


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


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


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


The truncated tetrahedron is an Archimedean solid made by 4 triangles and 4 hexagons.


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.


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.


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.


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: Sphere


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 study a kind of polyhedra inscribed in a sphere, in particular the Campanus' sphere that was very popular during the Renaissance.


Basic world map in a sphere. Latitude and longitude


This perpective projection is areapreserving. If we know the surface area of a sphere we can deduce the volume of a sphere, as Archimedes did.

Space Geometry: Rhombic Dodecahedron


Humankind has always been fascinated by how bees build their honeycombs. Kepler related honeycombs with a polyhedron called Rhombic Dodecahedron.


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?.


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.


You can build a Rhombic Dodecahedron adding six pyramids to a cube. This fact has several interesting consequences.


The Rhombic Dodecahedron fills the space without gaps.


A chain of six pyramids can be turned inwards to form a cube or turned outwards, placed over another cube to form the rhombic dodecahedron.

Space Geometry: Rhombicuboctahedron and pseudo rhombicuboctahedron


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.


This polyhedron is also called Elongated Square Gyrobicupola. It is similar to the Rhombicuboctahedron but it is less symmetric.


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


We can see the interior of the augmented rhombicuboctahedron. Luca Pacioli wrote that you 'can see the interior only with your imagination'.


Starting with a Rhombicubotahedron we can add pyramids over each face. The we get a beautiful polyhedron that it is like a star.

Plane developments of geometric bodies


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


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


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


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


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


Plane net of pyramids cut by an oblique plane.


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


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


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 .


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 .


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


Using cardboard you can draw plane nets and build polyhedra.


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!


A very simple technique to build complex and colorful polyhedra.


Simple technique to build polyhedra gluing discs made of cardboard or paper.


Italian designer Bruno Munari conceived 'Acona Biconbi' as a work of sculpture. It is also a beautiful game to play with colors and shapes.


With three golden rectangles you can build an icosahedron.


Modular Origami is a nice technique to build polyhedra.


Examples of polyhedra built using tubes.


Examples of polyhedra built using Zome.


Examples of polyhedra built using tensegrity.


Material for a session 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.


Material for a session 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.


Material for a session about polyhedra (Zaragoza, 23rd Octuber 2015) . Building a cube with cardboard and an origami octahedron.


Material for a session about polyhedra (Zaragoza, 21st October 2016). Instructions to build several geometric bodies.


Material for a session about polyhedra (Zaragoza, el 20th October 2017). Instruction to build an origami tetrahedron.


Microarquitectura is a construction game developed by Sara San Gregorio. You can play and build a lot of structures modelled on polyhedra.

Resources: Building polyhedra, 3d printing


Building tetraedra using 3d printing. The tetrahedron is a selfdual polyhedron. The center of a tetrahedron.


Building cubes and octahedra using 3d printing. Cube and Octahedron are dual polyhedra.

Real Analysis

Sequences and Series


Geometric sequences graphic representations. Sum of terms of a geometric sequence and geometric series.


One intuitive example of how to sum a geometric series. A geometric series of ratio less than 1 is convergent.


The geometric series of ratio 1/2 is convergent. We can represent this series using a rectangle and cut it in half successively. Here we use a rectangle such us all rectangles are similar.


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, the Euler's constant, is defined using a covergent series.

Powers and Polynomials


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 xintercept and the yintercept of a linear equation.


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


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


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


We can consider the polynomial function that passes through a series of points of the plane. This is an interpolation problem that is solved here using the Lagrange interpolating polynomial.

Rational Functions


Rational functions can be writen as the quotient of two polynomials. Linear rational functions are the simplest of this kind of functions.


When the denominator of a rational function has degree 2 the function can have two, one or none real singularities.


For large absolute values of x, some rational functions behave like an oblique straight line, we call this line an oblique or slant asymptote.


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


The derivative of a lineal function is a constant function.


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


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


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


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

Integral


The integral concept is associate to the concept of area. We began considering the area limited by the graph of a function and the xaxis between two vertical lines.


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


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


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


Archimedes show us in 'The Method' how to use the lever law to discover the area of a parabolic segment.


Studying the volume of a barrel, Kepler solved a problem about maxima in 1615.

Polynomial functions and integral


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


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


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

The Fundamental Theorem of Calculus


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


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

Piecewise Functions


As an introduction to Piecewise Linear Functions we study linear functions restricted to an open interval: their graphs are like segments.


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.


A continuous piecewise linear function is defined by several segments or rays connected, without jumps between them.


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


We can study several properties of exponential functions, their derivatives and an introduction to the number e.


Using the integral of the equilateral hyperbola we can define a new function that is the natural logarithm function.


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.


The main property of a logarithm function is that the logarithm of a product is the sum of the logarithms of the individual factors.


The logarithm of the number e is equal to 1. Using this definition of the number e we can approximate its value.


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.


After the definition of the natural logarithm function as an integral you can define the exponential function as the inverse function of the logarithm.


Different hyperbolas allow us to define different logarithms functions and their inversas, exponentials functions.


Mercator published his famous series for the Logarithm Function in 1668. Euler discovered a practical series to calculate.



Taylor's Polynomials


By increasing the degree, Taylor polynomial approximates the exponential function more and more.


Mercator published his famous series for the Logarithm Function in 1668. Euler discovered a practical series to calculate.


By increasing the degree, Taylor polynomial approximates the sine function more and more.


The function is not defined for values less than 1. Taylor polynomials about the origin approximates the function between 1 and 1.


The function has a singularity at 1. Taylor polynomials about the origin approximates the function between 1 and 1.


The function has a singularity at 1. Taylor polynomials about the origin approximates the function between 1 and 1.


This function has two real singularities at 1 and 1. Taylor polynomials approximate the function in an interval centered at the center of the series. Its radius is the distance to the nearest singularity.


This is a continuos function and has no real singularities. However, the Taylor series approximates the function only in an interval. To understand this behavior we should consider a complex function.

Complex Analysis

Product of complex numbers


We can see it as a dilatative rotation.


The multiplication by a complex number is a transformation of the complex plane: dilative rotation.


From a complex number we can obtain a geometric progression obtaining the powers of natural exponent (multiplying successively)

Complex Functions


Complex power functions with natural exponent have a zero (or root) of multiplicity n in the origin.


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


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


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


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


Podemos modificar las multiplicidades del cero y del polo de estas funciones sencillas.




Una primera aproximación a estas transformaciones. Representación de dos haces coaxiales de circunferencias ortogonales.


The Complex Exponential Function extends the Real Exponential Function to the complex plane.


The Complex Cosine Function extends the Real Cosine Function to the complex plane. It is a periodic function that shares several properties with his real ancestor.


The Complex Cosine Function maps horizontal lines to confocal ellipses.


Inversion is a plane transformation that transform straight lines and circles in straight lines and circles.


Inversion preserves the magnitud of angles but the sense is reversed. Orthogonal circles are mapped into orthogonal circles


The usual definition of a function is restrictive. We may broaden the definition of a function to allow f(z) to have many differente values for a single value of z. In this case f is called a manyvalued function or a multifunction.


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

Taylor's Polynomials


We will see how Taylor polynomials approximate the function inside its circle of convergence.


The complex exponential function is periodic. His power series converges everywhere in the complex plane.


The power series of the Cosine Function converges everywhere in the complex plane.

Probability

Random Variables


When modeling a situation where there are n independent trials with a constant probability p of success in each test we use a binomial distribution.


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


Poisson distribution is discrete (like the binomial) because the values that can take the random variable are natural numbers, although in the Poisson distribution all the possible cases are theoretically infinite.


The Normal distribution was studied by Gauss. This is a continuous random variable (the variable can take any real value). The density function is shaped like a bell.


One important property of normal distributions is that if we consider intervals centered on the mean and a certain extent proportional to the standard deviation, the probability of these intervals is constant regardless of the mean and standard deviation of the normal distribution considered.


The (cumulative) distribution function of a random variable X, evaluated at x, is the probability that X will take a value less than or equal to x. In this page we study the Normal Distribution.


Calculating probabilities of symmetric intervals around the mean of a normal distribution.


Student's tdistributions were studied by William Gosset(18761937) when working with small samples.



History

Pitagoras


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 show us in 'The Method' how to use the lever law to discover the area of a parabolic segment.


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


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

Leonardo da Vinci's drawings for Luca Pacioli's book 'De divina proportione'


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


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


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


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


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


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


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


We can see the interior of the augmented rhombicuboctahedron. Luca Pacioli wrote that you 'can see the interior only with your imagination'.

Durer


In his book 'Underweysung der Messung' Durer draw a nonregular pentagon with ruler and a fixed compass. It is a simple construction and a very good approximation of a regular pentagon.


He studied transformations of images, for example, faces.


Durer was the first who published in german a method to draw ellipses as cone sections.


Durer made a mistake when he explanined how to draw ellipses. We can prove, using only basic properties, that the ellipse has not an egg shape .

Kepler


Kepler used an intuitive infinitesimal approach to calculate the area of a circle.


Kepler was one mathematician who contributed to the origin of integral calculus. He used infinitesimal techniques for calculating areas and volumes.


Studying the volume of a barrel, Kepler solved a problem about maxima in 1615.


Kepler was one mathematician who contributed to the origin of integral calculus. He used infinitesimal techniques for calculating areas and volumes. In this page we study one optimization problem.

Cavalieri


Using Cavalieri's Principle we can calculate the volume of a sphere.

The Logarithm Function


Mercator published his famous series for the Logarithm Function in 1668. Euler discovered a practical series to calculate.
