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So let's go on in Lesson 6 and ask, well,

how can you explain those effects, those angular effects?

Seeing acute angles a little bigger than they really are, obtuse angles being

a little smaller than they really are, seeing these effects in and

tilt phenomenon, as being changed in the way they are.

How can you explain all that sort of phenomenology empirically?

And the answer is, in pretty much the same way that we did before.

So I come back to remind you here that when we looked and talked about straight

lines and collecting the experience that we've always had from straight lines,

we used laser range scanner to report the physical reality of the world asked

by applying a bunch of straight lines at different orientations.

And lengths to these scenes many times, millions of times, to get the statistical

distribution of line length in relation to what's really out there,

what's our frequency of occurrence over the human experience of line lengths.

Well, we can do exactly the same thing for angles.

All we have to do is make these applications angular instead of

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straight lines.

And that's what's been done here.

So if you look at these icons underneath these graphs,

and I'll explain these in detail in a minute.

These are the applications to this laser range scan scenes.

And exactly the same way we talked about before,

applying to the scenes now, not just a single straight line but

angles oriented in these different ways as indicated by

the series of icons underneath each of these two columns of panels.

So the upper three panels are natural scenes and

the lower two panels are carpentered scenes, scenes of human artifact.

And the reason for using those two categories or for looking at those two

categories, is that we want to make sure that there's no difference between

the intuitions that people have about carpentered scenes over the decades.

And there have been lots of them as I said before,

that intuition about corners going away from you and coming at you.

We will be sure that there's no difference between natural scenes and

carpentered scenes that could provide some kind of explanation along those lines.

And the answers you see from the similarity of these natural and

carpentered scenes is there really is no difference and

we can dismiss that kind of possibility as we did before,

just sort of generally that there's no intuitive explanation for these things.

Now, we can do it on the basis of actual data.

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So the reason for using different orientations is again,

you want to make sure that there's nothing special about the orientation

of the angles, that angle is presented in this way, this way, this way.

All give you more or less the same result.

And as you can see here, they do.

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So what's the result in each of these graphs

plotting the angle

from an acute angle, well not even an acute angle but a minimal acute angle,

a straight line, up through a right angle at 90 degrees.

Back down into an obtuse angle that disappears at 180 degrees.

So each one of these graphs is a test of angle orientations

going from acute to 90 degrees, and

then back from 90 degrees through a series of obtuse angles to 180 degrees.

And the y-axis on each of these graphs is the probability, the frequency of

occurrence determined by the laser brain scanning and how often you see angles in

these different orientations in whatever way you present them in natural scenes.

Or carbonate scenes and as I said there's no real difference between those.

So what's the result or what's the result showing here?

Well, in each of these graphs, you can see that they're all basically the same.

There is a much higher probability of occurrence of lines that are either

acute or obtuse than right angles, so again, the frequency or the currents

now showing here in this different graphical way, is very different.

We don't have the same frequency of occurrence of any angle and

the general rule is that acute and obtuse angles,

the more acute they become, the more obtuse they become, the more frequent

they become in our experience, and the least frequent is a right angle.

Well, why should that be?

What sense does that make?

Why should it be that our presentation from the real world of right angles

is less frequent than our presentation of either acute or obtuse angles?

Well, the answer again, I think, is not hard to see.

And let me try to explain it to you in terms of this diagram,

which is the reason.

The reason for seeing relatively few or relatively fewer projections

from the real world that are right angles than projections that are acute angles.

So here is the right angle, and these lines are the same length.

Each of these four black lines is an identical black line.

So these black lines are arranged to make a right angle,

these black lines are arranged to make acute or obtuse angles.

Either one these angles are acute, these angles are obtuse.

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The same length of lines as right angles are always going to take up

more space, more physical space than the lines,

the same length lines as acute or obtuse angles.

Why is that?

Well, again, this diagram just shows you why it is in common sense terms,

but in terms of, why it is in the world, you'll recognize that larger

planar surfaces in the world, are much less, frequent in occurrence.

So let's go back to this image, and just look at it.

So planer surfaces in the world that are large like this one

are going to be relatively rare compared to planer surfaces on these tree trunks,

on any of these leaves, etc.

These are going to be frequently occurring planer surfaces, but

these large planer surfaces are going to be occurring relatively rarely.

So coming back to this diagram, the reason why right angles

are in our experience less often occurring and

less often projected to the retinas than lines of the same

length that are forming acute or obtuse angles, is just for that simple reason.

Planar surfaces that are large are less frequent in the world

than planar surfaces that are small.

And because this planar surface, a smaller planar surface,

always is going to fit into a larger planar surface.

I mean, that's just common sense geometry.

You're going to get lots and lots of these compared to these.

You're going to get fewer of these right angle projections than acute or

obtuse angle projections, other things being equal.

So that's the reason for

seeing these angles differently.

The empirical reason is much the same as the reason for

seeing the line lengths differently,

it's just looking at angles in the same way that we looked at line lengths.

And again, there's not time in the course to show you all the ways of

explaining those different phenomena that I showed you for angles, but

the explanations are all fall out of this fundamental reason

based on the frequency of occurrence of angles in our human experience.