matematicas visuales visual math
Geometry

Morley triangle | matematicasVisuales Wallace-Simson lines | matematicasVisuales Wallace-Simson lines | Mostration | matematicasVisuales Steiner deltoid | matematicasVisuales Steiner deltoid is an 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 Dilative rotation | matematicasVisuales Durer | matematicasVisuales Los Embajadores de Holbein el Joven | matematicasVisuales
Equiangular spiral | matematicasVisuales Dilation and rotation in an equiangular spiral | matematicasVisuales The Diagonal of a Regular Pentagon and the Golden Ratio | 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 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 tetrahedronn | matematicasVisuales Regular dodecahedron | matematicasVisuales Volume of a regular dodecahedron (Flash version) | 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
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 Building polyhedra. Simple techniques (in Spanish) | matematicasVisuales


Triangles
Morley triangle | matematicasVisuales
If we trisect the angles of a triangle and consider the three intersection points of corresponding trisector lines we always get an equilateral triangle (Morley's triangle)
Wallace-Simson lines | matematicasVisuales
Each point in the circle circunscribed to a triangle give us a line (Wallace-Simson line)
Wallace-Simson lines | Mostration | matematicasVisuales
Interactive 'Mostration' of the Wallace-Simson line.
Steiner deltoid | matematicasVisuales
All the Wallace-Simson lines form a deltoid (Steiner Deltoid).
Steiner deltoid is an 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.

Plane Transformations
Dilative rotation | matematicasVisuales
A Dilative Rotation is a combination of a rotation an a dilatation from the same point.
Durer | matematicasVisuales
He studied transformations of images, for example, faces.
Los Embajadores de Holbein el Joven | matematicasVisuales
In this painting we can see, among lots of interesting things, an anamorphosis of a skull. (In Spanish)

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.

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'.
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.

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)
Sections in Howard Eves's tetrahedron | matematicasVisuales
Howard Eves, mathematician and historian of Mathematics, received the George Polya Award for the article Two Surprising Theorems on Cavalieri Congruence
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 tetrahedronn | matematicasVisuales
We show a sphere and the Howard Eves's tetrahedron with congruent sections.
Regular dodecahedron | matematicasVisuales
One eighth of a regular dodecahedon of edge 2 has the same volume as a dodecahedron of edge 1.
Volume of a regular dodecahedron (Flash version) | 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.

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.

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