matematicas visuales visual math

The first time I read about this amazing construction of a non-regular pentagon by Dürer was in Dan Pedoe's book 'Geometry and the Liberal Arts' (an Spanish version was published by Gustavo Gili Editores in 1979).

As Pedoe pointed out 'Dürer's interest in the construction of regular polygons is a reflection of their use in the Middle Ages in both Islamic and Gothic decoration and, after the invention of firearms, in the construction of fortifications.' (Pedoe, p. 68)

In Dürer's book 'Underweysung der Messung' ('Four Books of Measurement', published in 1525) the author gives an exact construction for a regular pentagon (taken from Ptolemy) but here we are playing with another construction of a pentagon, in this case is an equilateral but not equiangular pentagon.

Dürer thought that 'geometry is the right foundation of all painting' and he wanted to be accessible to painters, artists in general and craftsmen. Then he wrote his books in German and the his focus was practical. The construction that we are studying now is a good example because, although it is approximate, it is very accurate and simple to draw.

He draw it using a rusty compass (a compass with fixed radius) that makes the task easier. [Abu'l-Wafa Al-Buzjani (940-998), mathematician and astronomer, was interested in this kind of geometric constructions using a rusty compass and wrote 'A book on those geometric constructions which are necessary for a craftsman'].

Durer drawing of a non-regular pentagon:  Durer's Underweysung der Messung page with the drawing of two pentagons | matematicasVisuales
In this page of Dürer's book 'Underweysung der Messung' you can see two constructions of pentagons using ruler and compass: one is exact and the other is approximate.
Durer drawing of a non-regular pentagon, a trigonometry exercise:  Durer's construction of a non regular pentagon in his book Underweysung derMessung | matematicasVisuales
This is the approximate construction of an equilatera but not equiangular pentagon. (This construction was well known by craftsman in Durer's time and was publishen in the book 'Geometria Deutsch', a workshop manual, around 1484)

We already know that the five angles of a regular pentagon are 108º. Pedoe wrote that, in Dürer's approximate construction, the angles in the base are 108º21'58'', there are two angles less than 108º and the angle on the top is larger than 109º. This differences would hardly be detectable in a drawing.

We are going to use trigonometry to calculate these angles. For simplicity we can consider that que side length is 1.

Durer drawing of a non-regular pentagon, a trigonometry exercise:  the side lenght is 1 | matematicasVisuales

You can start drawing angle ABF:

Durer drawing of a non-regular pentagon, a trigonometry exercise:  angle ABF | matematicasVisuales

The distance between B and F is twice the height of an equilateral triangle:

Durer drawing of a non-regular pentagon, a trigonometry exercise:  The distance between B and F is twice the height of an equilateral triangle | matematicasVisuales

If you remember the properties of the central and inscribed angles in a circle, this is very easy:

Using the Law of Sines we can calculate angle FHB:

Durer drawing of a non-regular pentagon, a trigonometry exercise: angle FHB | matematicasVisuales

Now we can calculate angle ABH:

Durer drawing of a non-regular pentagon, a trigonometry exercise: angle ABH, this angle is bigger than 108º, but the error is very small | matematicasVisuales

And this angle is bigger than 108º, but the error is very small.

To calculate angles HKI and BHK we can start calculating BZ:

Durer drawing of a non-regular pentagon, a trigonometry exercise: distance between B and Z | matematicasVisuales

If M is the point in the middle between H and I:

Durer drawing of a non-regular pentagon, a trigonometry exercise: point M in the middle between H and I | matematicasVisuales

We can calculate angle MHK:

Durer drawing of a non-regular pentagon, a trigonometry exercise: angle MHK | matematicasVisuales

Then, the triangle IHK is an isosceles triangle and it is very easy to calculate angle HKI:

Durer drawing of a non-regular pentagon, a trigonometry exercise: angle HKI, This angle is bigger than 108º | matematicasVisuales

This angle is bigger than 108º.

Finishing our job is now an easy task. Remember that the five angles of a convex pentagon measure three times 180º. We can deduce angle BHK

Durer drawing of a non-regular pentagon, a trigonometry exercise: angle BHK, The angle at the top is smaller than 108º | matematicasVisuales

The angle at the top is smaller than 108º.

We can play with the zoom and see how accurate is Durer's approximation:

Even zooming you can see that the approximation is very good.

Durer drawing of a non-regular pentagon, a trigonometry exercise: very good approximation  | matematicasVisuales
Durer drawing of a non-regular pentagon, a trigonometry exercise: angle bigger than 108º  | matematicasVisuales

Very close we can see the small error:

Durer drawing of a non-regular pentagon, a trigonometry exercise: angle smaller than 108º, zooming we can see the small error | matematicasVisuales

The height of Dürer's pentagon is smaller than the height of a regular pentagon and the angle at the top is bigger:

Durer drawing of a non-regular pentagon, a trigonometry exercise: the angle at the top is smaller than 108º | matematicasVisuales

REFERENCES

Albrecht Durer's Underweysung der Messung in SLUB Dresden, Digital Collection.
Dan Pedoe, Geometry and the Visual Arts. Dover Publications. (pp. 66-73)
Donald W. Crowe, Albrecht Dürer and the regular pentagon, in Fivefolded Symmetry by Istvan Hargittai Ed., World Scientific Publishing Co.
Erwin Panofsky, The Life and Art of Albrecht Dürer.

MORE LINKS

The Diagonal of a Regular Pentagon and 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'.
The golden ratio
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.
Central and inscribed angles in a circle
Central angle in a circle is twice the angle inscribed in the circle.
Drawing fifteen degrees angles
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.
The golden rectangle
A golden rectangle is made of an square and another golden rectangle.
The golden rectangle and the dilative rotation
A golden rectangle is made of an square an another golden rectangle. These rectangles are related through an dilative rotation.
The golden spiral
The golden spiral is a good approximation of an equiangular spiral.
The golden rectangle and two equiangular spirals
Two equiangular spirals contains all vertices of golden rectangles.
The icosahedron and its volume
The twelve vertices of an icosahedron lie in three golden rectangles. Then we can calculate the volume of an icosahedron
Regular dodecahedron
Some properties of this platonic solid and how it is related to the golden ratio. Constructing dodecahedra using different techniques.
Plane developments of geometric bodies: Dodecahedron
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 .
Leonardo da Vinci: Drawing of a dodecahedron made to Luca Pacioli's 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.