How To Draw A Rectangle Using A Compass
Drawing a Regular Pentagon with ruler and compass
If we commencement with a segment we can draw a regular pentagon, only using ruler and compass, that has this segment as one side.
Nosotros already know that the diagonal of a regular pentagon are in golden ratio to its sides and that the golden ratio is denoted by 
This is the basic stride:
And then the value of a is:
And so we can describe a segment with length the diagonal of the regular pentagon:
And we can finish our work and become a regular pentagon:
Drawing twelve pentagons we get the net of a dodecahedron:
From Euclid'south definition of the division of a segment into its extreme and mean ratio we introduce a holding of gold rectangles and we deduce the equation and the value of the gilt ratio.
The diagonal of a regular pentagon are in golden ratio to its sides and the indicate of intersection of two diagonals of a regular pentagon are said to separate each other in the golden ratio or 'in extreme and mean ratio'.
More than LINKS
Demonstration of Pythagoras Theorem inspired in Euclid.
In his volume 'Underweysung der Messung' Durer draw a not-regular pentagon with ruler and a fixed compass. It is a simple construction and a very good approximation of a regular pentagon.
A aureate rectangle is made of an square and some other aureate rectangle.
A golden rectangle is made of an square an another gold rectangle. These rectangles are related through an dilative rotation.
The golden spiral is a proficient approximation of an equiangular spiral.
Ii equiangular spirals contains all vertices of aureate rectangles.
The twelve vertices of an icosahedron lie in three gold rectangles. Then we can calculate the book of an icosahedron
With three gilded rectangles yous can build an icosahedron.
Some properties of this platonic solid and how information technology is related to the golden ratio. Constructing dodecahedra using unlike techniques.
He studied transformations of images, for instance, faces.
Durer was the get-go who published in german a method to draw ellipses equally cone sections.
Durer made a mistake when he explanined how to describe ellipses. Nosotros can testify, using simply bones properties, that the ellipse has non an egg shape .
The starting time cartoon of a aeroplane net of a regular dodecahedron was published by D�rer in his book 'Underweysung der Messung' ('4 Books of Measurement'), published in 1525 .
Two transformations of an equiangular spiral with the aforementioned general efect.
In an equiangular spiral the angle between the position vector and the tangent is constant.
Leonardo da Vinci made several drawings of polyhedra for Luca Pacioli's book 'De divina proportione'. Here nosotros tin can see an adaptation of the dodecahedron.
A Dilative Rotation is a combination of a rotation an a dilatation from the same point.
Source: http://www.matematicasvisuales.com/english/html/geometry/goldenratio/drawingpentagon.html
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