Flatland, by Edwin Abbott Abbott

Flatland CoverToward the end of the 19th Century, physicists introduced the concept of a fourth spatial dimension (4D)  to the general population.   Try as you might, you cannot imagine what a 4D world would be like, but you can still make some conclusions about it by comparing our three dimensional (3D) world with a two dimensional (2D) world.

In 1884, Edwin A. Abbott wrote a delightful book that did just that.  His middle initial stands for Abbott.  His full name was Edwin Abbott Abbott that tells you something of his parents.  They were first cousins with the same last name, a fact they enshrined in the name of their son.  Edwin’s father was head of the Philological School in London.  Edwin took highest honors in mathematics and theology at Cambridge and became headmaster at the City of London School at age 26.

His unique book is called “Flatland” (that you can download free at ProjectGutenberg.com.)  Although his book has been used ever since to introduce physics students to the concept of extra dimensions, it is really a satire on the social regimentation of Victorian England, and once you catch on to this, it becomes hilarious, at least in the beginning.  It seems to me, Edwin started the book as a social satire but soon got caught up in his true love of geometry.

The narrator, a resident of a 2D world, “Flatland,” is telling us about his life there with no up-and down dimension—no height, depth, or thickness.  (He is speaking to us from prison for expressing the heresy of a third dimension.)

The Flatland people are of various geometric shapes based on their social status.  The more sides, the greater the status of the person.  Common laborers are isosceles triangles whose status is indicated by the spread of the narrow angle.  A laborer who reaches middle class, such as a merchant, grows into an equilateral triangle, but never becomes more than a triangle.  Triangles with points less than 10 degrees are so devoid of intelligence they become school teachers (his comment, not mine).

The narrator’s name is A. Square, indicating he is a professional man of the lowest rank but could become a pentagon someday.  When a rare person rises to the highest priestly status, he has gained so many sides he effectively becomes a circle.

Women have no status at all and are simply straight lines.  By law, they can only approach a man sideways because they are an almost invisible point when seen head on, and many a man has been stabbed by them, accidentally or on purpose.  For the safety of others, women have their own doors, which are very narrow.

Irregularity in any of the shapes indicates poor moral character and is carefully noted by the police.  A person is born with the irregularity, and it does not change over their lifetime.  (I warned you the book was a social commentary.)

Flatland houses and buildings look to us like floor plans on a sheet of paper, and we can watch the residents scurry about, even when they think they are hidden behind locked doors.  We can even see their internal organs.  Their buildings have no windows because light seems to be everywhere.  They cannot explain this, and those who try are heavily taxed.  Many Flatland worlds can be stacked together like a ream of paper, but each does not know the others exist, despite being only a tiny fraction of an inch away.

The Spaceland listener places the tip of his finger on a Flatland street.  The Flatlanders are amazed and run away in panic.  To them, a circle, the most holy of shapes, suddenly appears out of nowhere, then disappears just as mysteriously when the Spacelander removes it.

A. Square dreamed he visited Lineland, a one-dimensional world that is just one long line.  The Lineland inhabitants are segments on that line.  They cannot pass one another so their two neighbors are their neighbors forever, the only others they ever see.  They find this life perfectly normal and do not feel at all restricted.

In the same dream, he visits Pointland, a zero-dimensional universe that is only a single point inhabited by one, god-like resident, who also cannot imagine more dimensions and thinks his life, too, is perfectly normal.

So, here are just a few of the things we can infer about a fourth dimension:

  • Residents of a 4D world could see everything we do and even our internal organs, just as we can see into a Flatlander.  There is no place, and no thing, we could hide.
  • There may be many other 3D universes only a fraction of an inch away from us, but totally unknown, and perhaps forever unknowable.
  • Just as a sphere casts a circular 2D shadow, a 4D hypersphere projects a sphere.  In fact, any 3D object, even ourselves with all of our internal structure, could be only a projection of a 4D hyper-object
  • Any solid object could  suddenly appear in our world, seemingly out of thin air, by an intrusion from a 4D universe.
  • An additional large dimension could open vast intellectual growth and opportunities in our lives, but we cannot even imagine what they could be.

(In high school, I was fascinated by George Gamow’s popular and very readable book, One Two Three . . . Infinity, that goes into the science of all of this.  It is not on Project Gutenberg, but is well-known and available on Amazon and in libraries.)

Extra dimensions are now taken seriously in the forefront of physics.  Calculations of string theory vibrations match physical data if we assume the vibrations are in 13 dimensions, but physicists suspect the other 10 are tiny, subatomic size.  If this is true, then the real question is then why 3, and only 3, of our dimensions are so big.  Gravitational studies are hoping to detect another 3D universe inches away from ours, and there is serious speculation that our 3D universe (including us) is a holographic projection from the surface of a multidimensional hypersphere.   (See the posting, Time, and Where It Goes When It Goes By, October 02, 2009.)

Phew!  Fortunately, these speculations are beyond the scope of this blog.

RWalck@Verizon.net

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About Roger Walck

My reasons for writing this blog are spelled out in the posting of 10/1/2012, Montaigne's Essays. They are probably not what you think.
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One Response to Flatland, by Edwin Abbott Abbott

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