Thomas Knierim
27th March 2006, 10:36 AM
To The Inhabitants of SPACE IN GENERAL
And H. C. IN PARTICULAR
This Work is Dedicated
By a Humble Native of Flatland
In the Hope that
Even as he was Initiated into the Mysteries
Of THREE Dimensions
Having been previously conversant
With ONLY TWO
So the Citizens of that Celestial Region
May aspire yet higher and higher
To the Secrets of FOUR FIVE OR EVEN SIX Dimensions
Thereby contributing
To the Enlargement of THE IMAGINATION
And the possible Development
Of that most and excellent Gift of MODESTY
Among the Superior Races
Of SOLID HUMANITY
(Preface to the 2nd Edition of “Flatland” by Edwin A. Abbott, 1884)
If you envision yourself as a visitor to a two-dimensional world, as Edwin A. Abbott did, life would look quite different. Imagine a vast, presumably infinite surface teeming with lines, triangles, squares, circles, and more complex shapes and figures that, instead of remaining fixed in their places, roam freely, comprising the universe of a strange land called Flatland. I such a world you would be reduced, temporarily of course, to a conscious dot, not a geometrical point with zero dimensions, but rather a small unimpressive shape that dwells together with billions of other conscious shapes on a rather large plane, several trillion times bigger than the conscious shapes inhabiting it.
Being new to Flatland, you might expect to lay your eyes on a multitude of interesting shapes, polygons with varying symmetry features, vast regular and irregular appearances, and beautifully curved objects. In anticipation of these wonderful geometrical experiences, your disappointment is of course great, when upon arrival you suddenly realise that all you see is lines. In every direction you look there are lines and nothing but lines. These lines are moving and changing, appearing and disappearing, growing and shrinking; in fact, as you turn around, and look into each direction, Flatland looks like one big line to you.
Of course, you immediately realise your error. You remember that when you put a coin on the edge of a table, the coin appears as a circle only if you look at it from above and that, if you lower your eyes to the same height as the table surface, the coin will appear as a straight line. It dawns upon you that in Flatland there is no “above”, since height is a luxury of three dimensions. Thus, all shapes, whatever they might be, appear as lines to you.
Things are a little different from the Flatlanders’ perspective, however. The inhabitants of the plane see the same lines that you see, but have no difficulty in recognising and naming the shapes they encounter. This is even more surprising if you consider that they do this with only one eye at their frontal vertex. Because the Flatlanders see shapes from only one angle, they do not perceive the spatial displacements that we Spacelanders perceive. Given that they do not benefit from stereovision, you wonder how they recognise shapes at all. Edwin Abbott has the answer. Though the Flatlanders cannot see angles directly, they can infer them, and this with great precision. According to Abbott, fog makes this possible, or better, the two-dimensional equivalent of fog. If there were no fog, all lines would appear equally and indistinguishably clear; however thanks to fog, objects at a distance appear noticeably dimmer than objects that are close. The farther a line recedes into the background, the dimmer it becomes, and vice versa, the closer it is to the observer the brighter it appears. Presumably as a consequence of evolutionary adaptation, the Flatlanders are endowed with an extremely sensitive eye capable of distinguishing minute grades of comparative dimness and clearness, enabling them to infer with great precision the configuration of the object observed.
Consider this example. The Flatlander approaches a strange shape by bringing itself into a position where its glance bisects the angle a at vertex A of the approaching stranger. It chooses at fist the angle that is closest to its own seeing vertex. In case of a triangle, what is seen is the straight line DAE, in which the middle point A will be very bright because it is nearest to the observer; but on either side the line will shade away rapidly into dimness, because the sides DA and AE recede rapidly into the fog, and what appears as the triangle’s remote vertices, D and E, will be fairly dim. In case of an isosceles triangle seen from the top, DA and AE will appear equally long, and in case of an equilateral triangle, the Flatlander sees the same picture when moving around the shape and looking at its other vertices. In the case of a pentagon, the observation differs slightly. The line D'A'E' is seen with a bright centre A' and will shade away less rapidly to dimness, because the pentagon’s angle is wider than that of the former shape. The remote vertices of the pentagon, D' and E', will therefore not be as dim as the vertices D and E of the equilateral triangle. Rotation around the object again either confirms or denies the regularity of the shape.
From what was said, it is obvious that recognising a shape in the plane is not easy at all for the Flatlander. The process of recognition requires rotation, as well as the inference of its angles by looking at the degree of recession of each side into dimness. Because the Flatlander only sees a small part of the shape at any given time and because understanding the whole shape involves recombination of this information in memory, the Flatlander requires –besides good eyesight– solid training in geometry as well as a developed intellect to identify anything but the simplest shapes. The more complex a shape, the harder it is to recognise. The “divine intuition” that allows us humans to recognise a triangle, a square, or a circle in an instant is wholly beyond the imagination of the Flatlander.
Now, think of the amazement that befalls the Flatlander when coming into contact with a solid figure. Of course, Flatlanders have no conception of solidness, so we must ask how a three-dimensional figure appears to the Flatlander, or indeed if it does appear as such at all. Let’s assume a solid figure, say a sphere, comes into contact with Flatland and passes through the plane that the Flatlanders call their world. – What would the Flatlander observe? – At first, the Flatlander would observe a point at the location where the sphere touches the plane. The point then grows into a circle and continuously expands until it reaches the full diameter of the sphere, whereupon the motion ostensibly reverts, and the circle decreases in size until it becomes a point again and finally vanishes altogether. The average Flatlander would interpret this vision doubtlessly as a shape with magical capabilities. After all, we would think the same of a three-dimensional object capable of changing its form and becoming invisible at will.
Three-dimensional experience is as natural to us humans as two-dimensional experience is to the Flatlander. Space and time are so fundamental to us that we can hardly imagine anything without them. The notions of space and time are at the same time pervasive and elusive. Our everyday experience is filled with objects in motion that constantly reinforce our dynamic view of the world. Space and time are at the basis of physical theory and provide the framework for classical mechanics, which expounds the laws governing motion enabling us to rationally understand the dynamic world we see. If we abstract time and space from their particular object, they become objects themselves which maybe studied in their own right. This study then leads to fundamental questions about space and time, which are subject of physics and metaphysics.
For example, we ask whether space and time are finite or infinite, whether they are dependent on the object of perception, or dependent on the perceiver, or possibly even dependent on consciousness itself, and thus whether space and time exist at all outside human perception. Physicists rarely ask these types of questions, although they make use of the concepts of space and time very frequently. In physics, space and time are axiomatic, although there are subtle (and sometimes less subtle) differences in their interpretation, as they are most palpable when comparing classical mechanics to the 20th century view of space and time.
Cheers, Thomas
And H. C. IN PARTICULAR
This Work is Dedicated
By a Humble Native of Flatland
In the Hope that
Even as he was Initiated into the Mysteries
Of THREE Dimensions
Having been previously conversant
With ONLY TWO
So the Citizens of that Celestial Region
May aspire yet higher and higher
To the Secrets of FOUR FIVE OR EVEN SIX Dimensions
Thereby contributing
To the Enlargement of THE IMAGINATION
And the possible Development
Of that most and excellent Gift of MODESTY
Among the Superior Races
Of SOLID HUMANITY
(Preface to the 2nd Edition of “Flatland” by Edwin A. Abbott, 1884)
If you envision yourself as a visitor to a two-dimensional world, as Edwin A. Abbott did, life would look quite different. Imagine a vast, presumably infinite surface teeming with lines, triangles, squares, circles, and more complex shapes and figures that, instead of remaining fixed in their places, roam freely, comprising the universe of a strange land called Flatland. I such a world you would be reduced, temporarily of course, to a conscious dot, not a geometrical point with zero dimensions, but rather a small unimpressive shape that dwells together with billions of other conscious shapes on a rather large plane, several trillion times bigger than the conscious shapes inhabiting it.
Being new to Flatland, you might expect to lay your eyes on a multitude of interesting shapes, polygons with varying symmetry features, vast regular and irregular appearances, and beautifully curved objects. In anticipation of these wonderful geometrical experiences, your disappointment is of course great, when upon arrival you suddenly realise that all you see is lines. In every direction you look there are lines and nothing but lines. These lines are moving and changing, appearing and disappearing, growing and shrinking; in fact, as you turn around, and look into each direction, Flatland looks like one big line to you.
Of course, you immediately realise your error. You remember that when you put a coin on the edge of a table, the coin appears as a circle only if you look at it from above and that, if you lower your eyes to the same height as the table surface, the coin will appear as a straight line. It dawns upon you that in Flatland there is no “above”, since height is a luxury of three dimensions. Thus, all shapes, whatever they might be, appear as lines to you.
Things are a little different from the Flatlanders’ perspective, however. The inhabitants of the plane see the same lines that you see, but have no difficulty in recognising and naming the shapes they encounter. This is even more surprising if you consider that they do this with only one eye at their frontal vertex. Because the Flatlanders see shapes from only one angle, they do not perceive the spatial displacements that we Spacelanders perceive. Given that they do not benefit from stereovision, you wonder how they recognise shapes at all. Edwin Abbott has the answer. Though the Flatlanders cannot see angles directly, they can infer them, and this with great precision. According to Abbott, fog makes this possible, or better, the two-dimensional equivalent of fog. If there were no fog, all lines would appear equally and indistinguishably clear; however thanks to fog, objects at a distance appear noticeably dimmer than objects that are close. The farther a line recedes into the background, the dimmer it becomes, and vice versa, the closer it is to the observer the brighter it appears. Presumably as a consequence of evolutionary adaptation, the Flatlanders are endowed with an extremely sensitive eye capable of distinguishing minute grades of comparative dimness and clearness, enabling them to infer with great precision the configuration of the object observed.
Consider this example. The Flatlander approaches a strange shape by bringing itself into a position where its glance bisects the angle a at vertex A of the approaching stranger. It chooses at fist the angle that is closest to its own seeing vertex. In case of a triangle, what is seen is the straight line DAE, in which the middle point A will be very bright because it is nearest to the observer; but on either side the line will shade away rapidly into dimness, because the sides DA and AE recede rapidly into the fog, and what appears as the triangle’s remote vertices, D and E, will be fairly dim. In case of an isosceles triangle seen from the top, DA and AE will appear equally long, and in case of an equilateral triangle, the Flatlander sees the same picture when moving around the shape and looking at its other vertices. In the case of a pentagon, the observation differs slightly. The line D'A'E' is seen with a bright centre A' and will shade away less rapidly to dimness, because the pentagon’s angle is wider than that of the former shape. The remote vertices of the pentagon, D' and E', will therefore not be as dim as the vertices D and E of the equilateral triangle. Rotation around the object again either confirms or denies the regularity of the shape.
From what was said, it is obvious that recognising a shape in the plane is not easy at all for the Flatlander. The process of recognition requires rotation, as well as the inference of its angles by looking at the degree of recession of each side into dimness. Because the Flatlander only sees a small part of the shape at any given time and because understanding the whole shape involves recombination of this information in memory, the Flatlander requires –besides good eyesight– solid training in geometry as well as a developed intellect to identify anything but the simplest shapes. The more complex a shape, the harder it is to recognise. The “divine intuition” that allows us humans to recognise a triangle, a square, or a circle in an instant is wholly beyond the imagination of the Flatlander.
Now, think of the amazement that befalls the Flatlander when coming into contact with a solid figure. Of course, Flatlanders have no conception of solidness, so we must ask how a three-dimensional figure appears to the Flatlander, or indeed if it does appear as such at all. Let’s assume a solid figure, say a sphere, comes into contact with Flatland and passes through the plane that the Flatlanders call their world. – What would the Flatlander observe? – At first, the Flatlander would observe a point at the location where the sphere touches the plane. The point then grows into a circle and continuously expands until it reaches the full diameter of the sphere, whereupon the motion ostensibly reverts, and the circle decreases in size until it becomes a point again and finally vanishes altogether. The average Flatlander would interpret this vision doubtlessly as a shape with magical capabilities. After all, we would think the same of a three-dimensional object capable of changing its form and becoming invisible at will.
Three-dimensional experience is as natural to us humans as two-dimensional experience is to the Flatlander. Space and time are so fundamental to us that we can hardly imagine anything without them. The notions of space and time are at the same time pervasive and elusive. Our everyday experience is filled with objects in motion that constantly reinforce our dynamic view of the world. Space and time are at the basis of physical theory and provide the framework for classical mechanics, which expounds the laws governing motion enabling us to rationally understand the dynamic world we see. If we abstract time and space from their particular object, they become objects themselves which maybe studied in their own right. This study then leads to fundamental questions about space and time, which are subject of physics and metaphysics.
For example, we ask whether space and time are finite or infinite, whether they are dependent on the object of perception, or dependent on the perceiver, or possibly even dependent on consciousness itself, and thus whether space and time exist at all outside human perception. Physicists rarely ask these types of questions, although they make use of the concepts of space and time very frequently. In physics, space and time are axiomatic, although there are subtle (and sometimes less subtle) differences in their interpretation, as they are most palpable when comparing classical mechanics to the 20th century view of space and time.
Cheers, Thomas