matematicas visuales visual math
Geometry

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 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


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.

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.