Doing Math – Durer’s Polygons

Albrecht Durer was a 15th century artist and mathematician.  He started out his life as a gold smith who thought that adding mathematics to art would greatly improve it.  In the past goldsmith’s only had compass’s and straight edges in order to create complex polygons.  So Durer created his Four Books on Measurement which detailed how to make many different polygons with different numbers of sides.  Not all of his polygons are regular but he did at least come up with ways to make it close.  We will use his methods in order to work through constructing these polygons, though geogebra was used to create them the methods would work with pencil and paper.

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A regular quadrilateral is created by first using the compass and drawing a circle.  A horizontal line is then drawn through the center of the circle, the center is labeled A while the edges are labeled B and C.  We then draw a vertical line through A and perpendicular to line BC, we label these points D and E. We then connect D to B, B to E, E to C, and C to D.  We have now constructed a regular quadrilateral according to Durer’s method.

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We start the hexagon with a circle and a line drawn through the center.  We then create two more circles, with the same radius where the horizontal line intersects.  We place points where these two circles intersect the first one and then connect all of the points.

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To create a regular triangle (equilateral) we connect points CE, and F of the regular hexagon.

 

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In order to create and octagon we start with our regular quadrilateral and then create lines bisecting all of the sides of the quadrilateral.  Where these lines intersect the circle we place points and then we connect these points with those of the square and we create a regular octagon.

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To create a pentagon we first create horizontal and vertical lines through the center A.  We will then create a point E that is in the middle of A and B. We then form a circle with center E and a radius of ED. Where this circle intersects the horizontal will be point F.  We will take the length DF to be the length of the sides and then use this length to construct sides DG and DH We will use this same length for GI and HJ. We then connect points and Jthus completing our pentagon.

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