To do with shapes
Geometry is the branch of mathematics concerned with the
properties and relations of points, lines, surfaces and solids. The
majority of people probably don’t look back at the geometry they
learnt at school: the area of a circle, the lengths of the sides
of a right-angled triangle, the volumes of cubes and cylinders,
not forgetting those reliable tools of the trade, the compass and
protractor, with nostalgic fondness. I therefore hope that you are
not too put off by a chapter devoted to geometry.
In the spirit of this book’s crusade against the scientific
language of Jargonese, I will redefine the meaning of geometry
by saying that it has to do with shapes. Let us examine what we
mean by shapes in the most general sense. Look at the letter ‘S’. Its
shape is due to a single curved line. A splash of paint on a canvas
also has a shape, but this is no longer that of a line but an area.
Solid objects have shapes too. Cubes, spheres, people, cars all have
geometric shapes called volumes.
The property that is different in the above three cases—the
line, the surface and the volume—is the number of dimensions
required to define them. A line is said to be one-dimensional, or
1D for short, an area is two-dimensional, or 2D, and a volume is
3D.
Is there some reason why I could not go on to higher
dimensions? What is so special about the number three that
we have to stop there? The answer, of course, is that we live
in a universe which has three dimensions of space; we have the
freedom to move forward/backwards, left/right and up/down,
but it is impossible for us to point in a new direction which is
at right angles to the other three. In mathematics these three
directions in which we are free to move are called mutually
perpendicular, which is the mathematicians’wayof saying ‘at right
angles to each other’.
All solid objects around us are 3D. The book you are reading
has a certain height, width and thickness (all three quantities
being lengths measured in directions at right angles to each other).
Together, these three numbers define the book’s dimensions. In
fact, if you multiply the numbers together you obtain its volume.
This is not so obvious for all solid objects. A sphere, for instance,
needs only one number to define its size: its radius. But it is
still three-dimensional because it is a solid object embedded in 3D
space.
We see around us shapes that are either one-, two- or threedimensional,
never four-dimensional because such objects cannot
be accommodated in our three-dimensional space. In fact, we
cannot even imagine what a four-dimensional shape would look
like. To imagine something means building a mental model of it
in our brains which can only cope with up to three dimensions.
We would, quite literally, not be able to get our heads round a 4D
shape.
To many people, ‘one-dimensional’ means ‘in one direction’.
Adding another dimension to something means allowing it to
move in a new direction. True enough, but, you might ask, how
about that letter ‘S’? When writing an ‘S’ your pen traces curves in
different directions. How can the final shape still be 1D? Imagine
a dot called Fred that lives on a straight line (figure 1.1). Fred
is unable to move off the line and is restricted to movement up
or down it. We say that his motion is one-dimensional. In fact,
since the line is his entire universe, we say that Fred lives in a 1D
universe. But what if his universe were the letter ‘S’? How many
dimensions would he be living in now? The answer is still one.
He is still restricted to moving up or down the line. Granted, his
life may be more interesting now that he has a few bends to tackle,
but curving a shape does not increase its number of dimensions.
Another way of talking about the dimensions of a space is by
seeing how many numbers, called co-ordinates, we need to locate
a certain position within that space. The following example, which
I remember reading years ago but cannot remember where, is still
the clearest one I know. Imagine you are on a barge going down
a canal. Given some reference point, say that village you passed
earlier, you need just one number: the distance you have travelled
from the village, to define your position. If you then decide to stop
for lunch you can phone a friend and inform them that you are,
say, six miles upstream from the village. It doesn’t matter how
twisted the canal is, those six miles are the distance you travelled
and not ‘as the crow flies’. So we say that the barge is restricted to
motion in one dimension even though it is not strictly in a straight
line.What if you are on a ship on the ocean? You now require
two numbers (co-ordinates) to locate your position. These will be
the latitude and longitude with respect to some reference point,
say the nearest port or internationally fixed co-ordinates. The ship
therefore moves in two dimensions.
For a submarine, on the other hand, you need three numbers.
In addition to latitude and longitude you must also specify a length
in the thirddimension, its depth. Andsowesay that the submarine
is free to move in three-dimensional space.
Geometry is the branch of mathematics concerned with the
properties and relations of points, lines, surfaces and solids. The
majority of people probably don’t look back at the geometry they
learnt at school: the area of a circle, the lengths of the sides
of a right-angled triangle, the volumes of cubes and cylinders,
not forgetting those reliable tools of the trade, the compass and
protractor, with nostalgic fondness. I therefore hope that you are
not too put off by a chapter devoted to geometry.
In the spirit of this book’s crusade against the scientific
language of Jargonese, I will redefine the meaning of geometry
by saying that it has to do with shapes. Let us examine what we
mean by shapes in the most general sense. Look at the letter ‘S’. Its
shape is due to a single curved line. A splash of paint on a canvas
also has a shape, but this is no longer that of a line but an area.
Solid objects have shapes too. Cubes, spheres, people, cars all have
geometric shapes called volumes.
The property that is different in the above three cases—the
line, the surface and the volume—is the number of dimensions
required to define them. A line is said to be one-dimensional, or
1D for short, an area is two-dimensional, or 2D, and a volume is
3D.
Is there some reason why I could not go on to higher
dimensions? What is so special about the number three that
we have to stop there? The answer, of course, is that we live
in a universe which has three dimensions of space; we have the
freedom to move forward/backwards, left/right and up/down,
but it is impossible for us to point in a new direction which is
at right angles to the other three. In mathematics these three
directions in which we are free to move are called mutually
perpendicular, which is the mathematicians’wayof saying ‘at right
angles to each other’.
All solid objects around us are 3D. The book you are reading
has a certain height, width and thickness (all three quantities
being lengths measured in directions at right angles to each other).
Together, these three numbers define the book’s dimensions. In
fact, if you multiply the numbers together you obtain its volume.
This is not so obvious for all solid objects. A sphere, for instance,
needs only one number to define its size: its radius. But it is
still three-dimensional because it is a solid object embedded in 3D
space.
We see around us shapes that are either one-, two- or threedimensional,
never four-dimensional because such objects cannot
be accommodated in our three-dimensional space. In fact, we
cannot even imagine what a four-dimensional shape would look
like. To imagine something means building a mental model of it
in our brains which can only cope with up to three dimensions.
We would, quite literally, not be able to get our heads round a 4D
shape.
To many people, ‘one-dimensional’ means ‘in one direction’.
Adding another dimension to something means allowing it to
move in a new direction. True enough, but, you might ask, how
about that letter ‘S’? When writing an ‘S’ your pen traces curves in
different directions. How can the final shape still be 1D? Imagine
a dot called Fred that lives on a straight line (figure 1.1). Fred
is unable to move off the line and is restricted to movement up
or down it. We say that his motion is one-dimensional. In fact,
since the line is his entire universe, we say that Fred lives in a 1D
universe. But what if his universe were the letter ‘S’? How many
dimensions would he be living in now? The answer is still one.
He is still restricted to moving up or down the line. Granted, his
life may be more interesting now that he has a few bends to tackle,
but curving a shape does not increase its number of dimensions.
Another way of talking about the dimensions of a space is by
seeing how many numbers, called co-ordinates, we need to locate
a certain position within that space. The following example, which
I remember reading years ago but cannot remember where, is still
the clearest one I know. Imagine you are on a barge going down
a canal. Given some reference point, say that village you passed
earlier, you need just one number: the distance you have travelled
from the village, to define your position. If you then decide to stop
for lunch you can phone a friend and inform them that you are,
say, six miles upstream from the village. It doesn’t matter how
twisted the canal is, those six miles are the distance you travelled
and not ‘as the crow flies’. So we say that the barge is restricted to
motion in one dimension even though it is not strictly in a straight
line.What if you are on a ship on the ocean? You now require
two numbers (co-ordinates) to locate your position. These will be
the latitude and longitude with respect to some reference point,
say the nearest port or internationally fixed co-ordinates. The ship
therefore moves in two dimensions.
For a submarine, on the other hand, you need three numbers.
In addition to latitude and longitude you must also specify a length
in the thirddimension, its depth. Andsowesay that the submarine
is free to move in three-dimensional space.
No comments:
Post a Comment