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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Communicating Math – The Origins of Calculus

Calculus in the modern day has many uses and once a person has learned calculus it is hard to fathom how anything was built successfully without it.  Calculus in its simplest terms is the how quickly numbers change.  The two mathematicians credited with the creation of Calculus are Isaac Newton and Gottfried Leibniz, though it is argued that if Archimedes had lived a few more years that he would have created calculus a few centuries before either of them were even born.

Archimedes started studying how volume changes and how to calculate volume in a variety of different shapes.  Though without the use of calculus his proofs about volume are long and cumbersome.  But he did start to discover limits though he called it exhaustion and used it to approximate the area of circles by calculating the area of polygons, with roughly the same diameter, each with more edges than the previous iteration.

In the 17th century the idea of a derivative, the rate of change of a function, came into being. Newton and Leibniz are the main two mathematicians credited with creating calculus. While Newton was the first to come up with calculus, he used it to solve problems in physics, Leibniz was the first to publicize Calculus in its own right.  Newton did not publish any papers formalizing calculus instead he just used it in physics problems as can be seen in his correspondence.  To figure out the total area he used ratios of changes and showing that each change was infinitesimally small but the number of changes were also infinitesimal.

Leibniz on the other hand started his development of calculus by taking a graph and breaking it up into many small rectangles and then finding the area of these rectangles.  Leibniz made many of his discoveries by experimenting with the notation of calculus.  He is also who we can thank for the notation behind calculus.

Calculus has been centuries in the making and its development has had contributions from many intelligent mathematicians.  In its simplest terms it is the calculations behind the rate of change.

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