**The Vitruvian Man and the Squaring of the Circle**

by Klaus Schröer

**The Vitruvian Man and the Squaring of the Circle**

This presentation is based on the German book

"Ich aber quadriere den Kreis ..." by Klaus Schröer and Klaus Irle

**The Vitruvian Man and the Squaring of the Circle 1/1**

In 1492 Leonardo da Vinci created the most famous drawing ever:

the so-called Vitruvian Man.

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On the paper he drew a male figure with horizontal arms and vertical legs...

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...and another one with arms and legs stretched out.

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He combined both to a double figure, which illustrates a movement of legs and arms.

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Moreover, the first figure is marking a square...

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...and the second one is marking a circle.

The general idea of a man with ideal proportions describing a square and a circle wasn't Leonardo's own.

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This idea was first described by the ancient Roman architect Vitruv.

And Leonardo gave credit to him by quoting this text in his own words at the top and the bottom of the paper.

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Moreover, he put a scale under the drawing which is referring to the special proportions of the human body that Vitruv wrote about.

Last but not least, he signed the paper and some catalogue notes were added later.

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But this drawing contains a fascinating secret:

It can be seen as a geometrical algorithm for the so called squaring of the circle in an infinite number of steps.

Before going into this in detail, let us have a look what the squaring of the circle is about.

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It is a geometrical problem having been raised by the ancient Greeks.

The goal is to construct a pair of a square and a circle in such a way...

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...that the area of the square is equal...

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...to the area of the circle...

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...only using two tools: a circle and a scaleless ruler.

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Over the centuries many people tried to solve this problem. They all failed, until the German mathematician Lindemann

was able to prove in 1823 that this problem can only be solved in an infinite number of construction steps.

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Well, did Leonardo knew about Lindemanns proof?

Of cause not.

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Was Leonardo interested in the problem of squaring the circle?

He was obsessed by it...

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...and he claimed to have a solution in an infinite number of steps for it!

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Now let us come to the algorithm of the Vitruvian Man.

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The key to understand Leonardo's drawing is to add another circle...

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...which has the same area as the square Leonardo showed us...

**The Vitruvian Man and the Squaring of the Circle 3/4**

...and to add another square...

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...which has the same area as the circle Leonardo showed us.

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This results in two pairs of a square and a circle - a smaller pair and a bigger pair - and, as we will see now,

these two pairs are deeply connected by geometric constructions.

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The bigger circle is a construction result shown by the movement of the arms based on the smaller pair of a square and a circle.

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Leonardo has drawn in the two center points of these arm movements and thus one obtains two construction circles which represent the movement of the arms.

They lead us from the points of intersection of the smaller pair of square and circle to the upper points of intersection of the new bigger circle and the smaller square.

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Together with the point this circle is standing on, we have three points of its circumference which enables us to construct it

including its center point, which is exactly the man's navel in the figure.

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The bigger square is also a construction result, as shown in this animation:

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What you need here is just the center of the big circle we just have constructed and

the lower corner points of the small square.

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Then draw two lines from the corner points through the center of the big circle until you hit the outline of the bigger circle again,

and take the height of these points of intersection as the size of the new square.

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So now we have a complete construction, leading from the smaller pair of square and circle to the bigger one.

Before going on just watch the complete animation of this way of construction twice and slowly.

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To have an unusual way to construct an equal bigger pair of a square and a circle is nice, but clearly not a way to square the circle

since the construction starts with an equal pair. So, what's the deal?

**The Vitruvian Man and the Squaring of the Circle 4/1**

The deeper meaning of this construction only becomes clear when it is carried out on an unequal starting pair.

In this example the area of the circle has only 90% of the area of the square.

**The Vitruvian Man and the Squaring of the Circle 4/2**

Imagine Leonardo's figure with the horizontal arms here. It would look very different.

But I didn't illustrate this here, since I would like to avoid injuring anybody because of his/her body proportions. This is not my intention.

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Instead we will now construct a new pair of a square and a circle from this pair

by using what we have learned from Leonardo's drawing.

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And for this experiment we will use the same radius of the arm movements

in relation to the edge length of the starting square Leonardo used.

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So in this animation the construction of the new pair is shown. The animation starts showing the starting pair,

but in the end you will only see the resulting one.

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And here we go! Starting with a pair with 10% difference between both areas

the new pair we have just constructed has only a difference of about 1.6%.

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Now we will add a figure with horizontal arms representing this pair.

As we will see in the next picture, the proportions of this figure are already very close to those in Leonardo's drawing.

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Let us compare in the next slide the result with the original drawing.

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The difference is difficult to see. OK. Was the dramatic change in the area ratios just a coincidence?

What happens if we try other ratios for the areas of the starting pairs?

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Well, a lot of experiments were done on these questions.

At first they were carried out in the old-fashioned way: with a paper, a circle and a ruler and by measurement.

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Because of the amazing results of these first attempts, the construction was mathematically modeled as you see here ...

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...and translated into a simple javascript that simulates the construction computationally.

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Both the mathematical modeling and the javascript are available for download

on the website www.klaus-schroeer.com/leonardo/ (Look for "Resources" at the bottom of the page).

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Now let's come to the amazing results the construction produces.

On the y-axis on the left we see different area ratios for the starting pairs between 0.8 and 1.25...

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...and one step to the right we see the area ratios of the resulting pairs produced by the construction.

As you can see, the resulting pairs all seem to be closer and closer to a ratio of 1 - the squaring of the circle!

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And now imagine you had been Leonardo and you had just figured this out. What would be the obvious next step?

Yes: To carry out the construction on the result of the construction again!

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The result is indeed mind-blowing. In the third generation all ratios of the areas of squares and circles are extremly close to 1.

This operation was not meant to be carried out just once. No.

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The algorithm of the Vitruvian man was and is meant to be carried out repeatedly, going on and on forever and thus

yielding a recursively defined infinite sequence.

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Using the value for the radius of the arm movements, Leonardo showed us in the Vitruvian Man

the area ratios of the sequence of pairs of squares and circles come clother and clother to a value of 1.00037.

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

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OK. 1.00037 is not the ideal squaring of the circle, but it is the best result of the Renaissance - way better then Dürer or Luca Pacioli, for example.

But before we come to a conclusion, we have to examine one last point:

**The Vitruvian Man and the Squaring of the Circle 5/1**

Denote by x the radius of the arm movements, which is the distance between these two points here.

Up to this point we have only used the radius Leonardo gave us in his drawing which was 0.436 and produced

circles and squares approaching to a ratio of areas of about 1.00037.

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Now we will trie other possibilities for the radius x

in an area between 0.5 ...

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and 0.4...

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...and in steps with a length of 0.01 between those points as you see here.

And we will plot the results in here by using this vertical scale for the area ratios obtained from different values of x.

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Here is then the result. The value for x Leonardo used is marked on the plot. OK. What can we learn from this plot?

At first you don't have to care about the value of x doing the construction for practical purposes.

We get excellent results all over the intervall between 0.5 and 0.4.

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But more important:

This plot looks like a function that is semi-continuous and strictly monotonously increasing.

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And that means: there must exist, in precise mathematical sense, a value of x that produces pairs of squares and circles with equal areas

somewhere between 0.45 and 0.46 and marked here by an arrow.

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So let's now calculate this x. The trick to do that is this: You assume that you already have this radius x and

that the recursively defined infinite sequence has already reached its limit. Then two consecutive pairs of a square and a circle must both already be exactly equal in area.

This is enough to calculate the value of this ideal x. Here is the calculation:

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The result is this. Since we are dealing with the squaring of the circle, it should also not be too bewildering to see that it is a related to the famous transcendental number Pi = 3,14...

We can also give a numeric representation of x: it's about 0.4535605...

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And because it is a formula explicitly involving Pi you also need an infinitive number of construction steps to build this thing with a circle and a scaleless ruler.

But Leonardos algorithm works in an infinite number of steps.

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So maybe it is possible to do this in the following way: you start, let's say, with x = 0.5 and then you construct a new x

in every generation with a finite number of construction steps. And this x will converge to the value we see here!

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And imagine that the construction steps for this ideal x could also be considered as an expression of the proportions and movements of the human body.

Modern mathematics does not rule out this possibility!

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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.

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This limit approaching behaviour of the ratios takes place very quickly and is, in mathematics, 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.

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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 almost equal areas.

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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 the ancient Roman architect Vitruv.

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And, indeed, it can't be ruled out, that there is an ideal algorithm based on Leonardos ideas

which does the job of squaring the circle perfectly in an infinte number of steps and which might also to be found in the human movements and proportions.

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