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+++ There will be further research on the algorithm of the Vitruvian Man at the Department of Mathematics of the KIT (Karlsruhe Institute of Technology) in 2019 (500th anniversary of Leonardo's death). +++                                                                                                                                                                              

This website is based on the German book

"Ich aber quadriere den Kreis ..."

Leonardo da Vincis Proportionsstudie

Prospekt

by Klaus Schröer, Klaus Irle

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Results


The Vitruvian Man by Leonardo da Vinci depicts a geometric construction producing a recursively defined infinite sequence of pairs of squares and circles.

Starting this sequence with a pair with unequal areas, the ratios of the areas of the following pairs come closer and closer to a value of 1.00037 by every generation.

This behaviour of the ratios takes place very quickly and is called convergence.

This sequence was meant as a solution of the so-called squaring of the circle in an infinite number of steps, that Leonardo claimed to possess.

In his drawing Leonardo exhibited the construction and the sequence on the basis of two successive generations - generation n and generation n+1 - and already in the state of convergence, because both pairs have already equal areas.

And he only showed us the square of generation n and the circle of generation n+1.

The most astonishing fact is that Leonardo discovered this algorithm in the proportions and movements of the human body based on the ideas of ancient architect Vitruv.

Animations


See the algorithm at work

YouTube Documentation

Leonardo's Vitruvian Man

and the Squaring of the Circle

by Klaus Schröer

Feel free to embed this video on your website:

Video Link

Or watch the story step by step with text

(attention: the page needs one minute to load)

Prospekt

Leonardo's Vitruvian Man and the Squaring of the Circle - step by step

Resources


to experiment and teach

The math
(PDF German)

The math behind the Vitruvian Man. The pdf contains the modeling of the construction from generation n to n + 1, the calculation of the ideal radius of the arm movement x, a listing of the simple javascript the calculations were done with and listings of the calculation results.

Javascript
simulation

This is the extreme simple javascript the calculations for the book were done with. It calculates the first ten generations. To change the value of x (x[1] in line 18) or the ratio of the areas of the starting pair (identical to the area of the first circle f[1] in line 19, because the area of the first square is defined as 1) you have to edit the source code. You can use a simple text editor for that.

Javascript
with graphics

This more advanced script makes a drawing of the complete construction based on the ratio of the areas of the starting pair that you define. Again the ratio of the areas of the starting pair is identical to the area of the first circle of the starting pair, because the area of the first square is defined as 1. Please follow the brief instructions for the script to work.

Pictures
and animations

Here you will find commented pictures and animations. They can be used for educational and scientific purposes. Please note the information regarding use for other purposes. The unaltered illustration of the Vitruvian Man can be used freely.

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