Here I will use some simple examples from geometry that anyone, artist or scientist, can visualise, to show how important Visual Thinking is to everyone.

Hopefully, you will discover that even quite difficult aspects of geometry are accessible to everyone. Geometry, I suggest, is where artists and scientists meet on common ground and Visual Thinking is their common practice.

## Introduction

Much of our ability to reason is based on our ability to hold images in our minds. This is the place where the artist and the scientist share a common heritage. In visual art, creating an image requires the artist to first form an idea, to visualise what it is they plan to create. In science, the idea is usually a hypothesis or conjecture to be tested, also visualised as an image in the scientist's mind.Perhaps the simplest form of visual image, and the most common, is that of a geometric structure is space, a square or a block for example. These shapes are all around is in our everyday experience and comprise a language, Geometry, common to both art and science. Reasoning about shapes is something that we are all equipped to do from our everyday experience.

Geometry was one of the components of a classical liberal education. Geometry was important primarily because it was a clear way of introducing logic and reasoning in a context that everyone could be expected to understand.

A significant aspect of this ability to understand Geometry is because the subject matter, structures in space or on a plane, is something we can all easily visualise. Geometry engages our ability to visualise simple physical structures. It engages our Visual Thinking.

## An first example of Visual Thinking

As a first example of Visual Thinking, I am going to ask you to construct, in your head, a simple argument about triangles and circles.I would prefer you to try to do this only by visualising the structures about which we are reasoning, rather than by looking at a diagram. Consequently, I won't provide a diagram myself. If it helps, however, there is no harm in you sketching a diagram as you read about the structures I want you to visualise. Sketching can be a significant aid to thinking.

Let's start. Imagine a triangle with a circle inside it. This circle is drawn in such a way that it exactly touches all three sides of the triangle. There is no gap between the circle and a side. Nor does the circle cross a side twice, by venturing slightly over the side before turning back into the triangle. The circle makes three glancing touches with the triangle, one touch with each side.

Can you visualise that? If not, read the paragraph again, perhaps making some sketches until the paragraph makes sense.

Now comes the reasoning part. Of all the circles we could draw touching the sides of a triangle, there is only one circle that will exactly touch all three sides. We can't make a larger or smaller circle fit into the triangle, no matter how we move the circle around. The only way to make a larger or smaller circle fit would be to alter the triangle.

If we assume the triangle is fixed, there is only one circle that fits exactly inside it. Can we prove that simply by thinking about it? Could we use our visual thinking to construct a verbal argument that would engage a listener or reader's ability to visualise our proof? Let's try.

Imagine the triangle. For the moment, think only of two of its sides. Imagine drawing a circle which exactly touches both these sides, fitting into the vee that they form. Since we have been ignoring the third side, this circle will probably be one that doesn't touch that third side. It will be too small, or it will be too big.

Now add that forgotten third side back into the image you have of two sides and a circle in their vee. If we allow the circle to grow or shrink there will come point when that circle exactly touches the third side. There will only be one such point. There will only be one such circle.

For me this argument proves that the circle is unique. That, given a triangle, there is only one circle that fits exactly inside it. The supporting argument is one we have built in our heads. One we could repeat to convince a friend. One that sits neatly in the place where art and science meet.

## A second example of Visual Thinking

Interestingly, we must arrive at exactly the same circle no matter which two sides we choose to begin our construction. Can you see that?We can use that idea to determine a way to draw the circle exactly fitting inside a given triangle using only a compass and a straight edge. That is what we will do next. Maybe you would like to try thinking about it yourself before going on.

In plane geometry it is common practice to consider how to construct a particular shape using only a straight edge and a compass.

The straight edge (usually a ruler) is used to draw precise straight lines between two points on a page. It is called a straight edge because we are not to use it for any purpose other than drawing straight lines. In paricular, we are not to use it for measurement.

The compass is a tool that allows us to draw circles but is particularly reserved for making exact copies of lengths already constructed on the page. We do this by spanning a given length, between to points on the page, with the two points of the compass (the pin and the pencil) and then moving the compass to a new location on the page marking off a copy of the length we just collected.

I won't ask you here to make compass and straight edge constructions yourself, unless you wish to, but rather ask you to imagine doing so as a further exercise in visual thinking.

For this exercise I need you to accept that it is possible, using just compass and straight edge, to divide an angle precisely into two parts. This is accomplished by drawing a third line through the point where two lines meet, such that the third line breaks the angle between to first two lines precisely down the middle.

I need you to accept that, or to convince yourselves of it without my help. You need to be able to visualise that third line that splits a given angle into two equal parts, that goes right down the middle between the two lines forming the angle to be halved.

Now comes the visual thinking bit. The third line through the middle has the propery that any circle that would exactly touch the first two lines will have its centre on that third line.

Do you see that? It's because a circle touching two lines will be equally distant from those two lines. All the points on that third line are equally distant from the first two, so they are the centres of touching circles.

Let's think again of our triangle with its exactly fitting circle. The centre of this circle will lie on the lines which exactly halve the angles of the triangle. We claim to already know how to construct such lines using compass and straight edge, so let's imagine constructing two such lines for any two angles of the triangle. Where these two lines meet will be the centre of our exactly fitting circle. Do you agree?

So we know where to place our compass if we want to actually draw that exactly fitting circle. But how do we know how wide to set our compass? This is where the artist and the scientist might part company. In practice, if we are only intent on actually drawing the circle, and we are using a normal pencil on normally rough paper, estimating the size of the circle be stretching the compass between the recently constructed centre for the circle and any one of the triangle sides it must touch would be perfectly sufficient. We draw the circle and anyone looking at it (especially if it has eventually been filled with paint) would think it was fitting exactly.

But a scientist would know that that estimate of the circle's size could be subject to a slight error caused by our problematic human visual capacity. The scientist would want to go on and construct, using just compass and straight edge, the exect length that is the size of the circle. Can you see how to do that? I won't go through that detail here. Rather, I hope that the visual thinking exercise you have just completed is sufficient to support my argumet that Visual Thinking is a practice familiar to both artist and scientist and that Geometry is one place where both artist and scientist meet.

## A third example of Visual Thinking

Now let's step up a gear. Let's really test our ability to visualise and to reason using Visual Thinking. Let's draw our triangles and circles on the surface of a sphere instead of on a flat sheet. If you stick with it, you will see this leads to a very unusual idea.Again, I hope this is an exercise you can do completely in your head, without the aid of a diagram, so I won't provide a diagram. That's not to say you shouldn't try drawing some helpful sketches yourselves if it helps. Once you've got the idea however, you will find that you can just run through it in your head (for example, when you are walking) without the aid of a diagram.

So, imagine drawing a triangle on the surface of a sphere. Geometry does have a precise definition of what that means. Because lines are no longer straight, curving as they do to stay on the surface, we need to be careful about what we call a straight line. What does carry over from plane geometry is that a line on a sphere is straight if it is the shortest distance between two points. You can imagine it as the position that would be taken by a piece of elastic stretched between those points.

Three points on a sphere would naturally give us the corners of a triangle. We could construct that triangle be stretching lines between each pair of points. On a large enough sphere, it wouldn't look too different from the plane triangle with which we are familiar.

Now we can also draw a circle inside this triangle that exactly fits by just touching each side. This circle will also be on the surface of the sphere. Every point on that circle will be equally distant from one point on the sphere which we can call the centre of the circle. The arguments we have given for the circle being unique and for constructing the circle on a plane carry over directly to this geometry on the surface of a sphere. Can you see that? Can you see the structures in your visual mind?

Something has changed, however. Something very interesting. Unlike the plane, which is infinite in all diections, the surface of a sphere is finite. On a plane we can draw ever bigger triangles. We can expand a triangle as much as we like. We just need more and more paper.

On a sphere, as we imagine our triangle getting bigger, eventually it reaches a size where bits of it begin to disappear over the horizon.

What's happening? Visualise that triangle on the sphere where the sides of the triangle are elastic. Allow the corners of the triangle to move apart so that the triangle slowly gets bigger.

Eventually the triangle will reach a point where its sides no longer get longer. Where they start to get shorter, or at least one of them does. The triangle has reached its maximum extent.

As the triangle continues its journey of expansiion, an independent observer would be forgiven for thinking that its actually a small triangle becoming smaller. Because what was the inside of the triangle now looks more like the outside.

And vice-versa.

So, thinking about that original triangle drawn on the surface of the sphere, which was the inside and which was the outside?

We were inclined to thing that the smaller of the two areas enclosed by the sides was the inside. But there is no logical basis for that. That's the unusual thing about geometry on the surface of the sphere. The triangle on the surface actually separated two distict areas, either of which we could have called the inside, calling the other the outside.

But what if we'd called the larger area the inside? Could be have constructed a circle on the surface of the sphere which fitted inside the triangle and exactly touched all three sides. So that the circle would now seem to be outside, rather than inside.

The answer is, we could construct such a circle. We could use exactly the construction we learned for constructing the circle inside a plane triangle. We would construct its centre by halving the angles and then construct the circle by drawing a curve that touched the triangle and remained at a fixed distance from the centre. So we would definitely end up with a circle.

Oddly, it would be exactly the same circle as if we'd chosen the smaller area as the inside. Isn't that remarkable? If you have managed to construct that in your head, with or without sketches on paper, you will now have a warm feeling of enlightenment. Welcome to Visual Thinking.