Children's Health & Safety Association

Issue 43: July 2018

Wednesday, 30 April 2014 18:04

Non-Euclidean Geometry

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April 30, 2014 – “In the night I had a dream. I saw before me a vast multitude of small Straight Lines (which I naturally assumed to be Women) interspersed with other Beings still smaller and of the nature of lustrous points - all moving to and fro in one and the same Straight Line, and, as nearly as I could judge, with the same velocity.”


This passage is taken from the first paragraph of “Part II: Other Worlds” in the 1884 book “Flatland: A Romance of Many Dimensions” by Edwin Abbott (1838 – 1926). The author created a fictional, two-dimensional universe called ‘Flatland’ (inhabited by lines, triangles, squares, and other geometric figures which only have ‘length’ and ‘width’, no ‘height’), in order to poke satirical fun at various elements of Victorian society. In “Part I: This World” Abbott describes the social order of his universe, but Part II is used to examine the possibility of multiple dimensions.

The principal character is a Square, a humble inhabitant of the world, who postulates on the nature of the universe. In a dream, he sees a one-dimensional universe – Lineland – that has only ‘length’ (no width or height). Its inhabitants are line segments which are aligned in a row along this Line.



They may move, but only “northward” or “southward” in direction; they cannot move past each other. They do communicate by talking, so Square, who cannot be seen by the inhabitants of this simple world, attempts to convince the king that he is from a two-dimensional universe. Part of the dialogue is given here:


King: “And let me ask what you mean by those words 'left' and 'right.' I suppose it is your way of saying Northward and Southward."

Square: "Not so," replied I; "besides your motion of Northward and Southward, there is another motion which I call from right to left."

King: “Exhibit to me, if you please, this motion from left to right.”

Square: “Nay, that I cannot do, unless you could step out of your Line altogether.”

King: “Out of my Line? Do you mean out of the world? Out of Space?”

Square: “Well, yes. Out of your World. Out of your Space. For your Space is not the true Space. True Space is a Plane; but your Space is only a Line.”

King: “If you cannot indicate this motion from left to right by yourself moving in it, then I beg you to describe it to me in words.”

Square: “If you cannot tell your right side from your left, I fear that no words of mine can make my meaning clear to you. But surely you cannot be ignorant of so simple a distinction.”

King: “I do not in the least understand you.”

The irony of this situation occurs when Square himself is visited by a being, called a Sphere, from a three-dimensional universe. Sphere, who cannot be seen by Square, tries to convince him that the true universe has length, width, and height. Square is unable to comprehend such a thing.

Mathematics plays a key role in the book. Consider this discussion that Square has with one of his grandsons, “a most promising young Hexagon of unusual brilliancy and perfect angularity.”

Hexagon: "You have been teaching me to raise numbers to the third power. I suppose 33 must mean something in Geometry; what does it mean?"

Square: "Nothing at all, not at least in Geometry; for Geometry has only Two Dimensions."

And then I began to shew the boy how a Point by moving through a length of three inches makes a Line of three inches, which may be represented by 3; and how a Line of three inches, moving parallel to itself through a length of three inches, makes a Square of three inches every way, which may be represented by 32.

Upon this, my Grandson, again returning to his former suggestion, took me up rather suddenly and exclaimed…


Hexagon: "Well, then, if a Point by moving three inches, makes a Line of three inches represented by 3; and if a straight Line of three inches, moving parallel to itself, makes a Square of three inches every way, represented by 32 ; it must be that a Square of three inches every way, moving somehow parallel to itself (but I don't see how) must make Something else (but I don't see what) of three inches every way - and this must be represented by 33."

Square: "Go to bed," said I, a little ruffled by this interruption: "if you would talk less nonsense, you would remember more sense."

As three-dimensional beings, humans can understand the mental challenge put forth by the grandson. If a 3 by 3 square (with an area of 3X3= 32 =9 square units), lying flat, was raised up 3 units, it would trace out a shape called a cube. The cube would have six sides. Each side would be a 3 by 3 square. This cube would have a volume of 3X3X3= 33 =27 cubic units.

Sphere, in fact, did try to make Square think about this strange concept by talking about an “upward” movement, something different from the side-to-side movements that occur in Flatland.

It is simple for a three-dimensional being to visualize one-dimensional and two-dimensional models. It is not easy for a two-dimensional being like Square to visualize the concept of three dimensions, whether the challenge is put forth by a young child called Hexagon or by a visiting alien called Sphere.

Sphere asked Square to consider the following relationships:












2 (bounded by 2 points)




4 (bounded by 4 lines)




6 (bounded by 6 squares)

The numbers in the chart become more interesting when presented in another manner. The “number of vertices” may be expressed as powers. A ‘power’ is a number (called the ‘base’) raised to an ‘exponent’. If the exponent is a natural number (such as 1,2,3,4, etc.) then the power can be described in this fashion: the natural number exponent indicates the number of times that the base must appear in a multiplication expression. For example, 5² = 5x5 =25  and 34 = 3x3x3x3 = 81 . If the exponent is a zero, the meaning of any base raised to the exponent 0 is 1. Thus, 50 = 1 and 30 = 1.

Consider the chart above rewritten in this fashion:







 20 = 1  0x2 = 0



 2¹ = 2

1x2 = 2 (bounded by 2 points)



 2² = 4

2x2 = 4 (bounded by 4 lines)



 2³ = 8

3x2 = 6 (bounded by 6 squares)

The third column numbers are all powers of 2.

The fourth column numbers are all multiples of 2.

In the book, Square was magically transported into the Third Dimension so that he could finally observe the truth of Sphere’s universe. Once his eyes had been opened to the new reality, he began to imagine greater things – and the interesting relationships in this table made him speculate on further extensions.

Square: “And consequently does it not of necessity follow that the more divine offspring of the divine Cube in the Land of Four Dimensions, must have 8 bounding Cubes: and is not this also, as my Lord has taught me to believe, "strictly according to Analogy"?

The following extension to the previous table illustrates Square’s thinking.






Hypercube (Tesseract)


 24 = 16

4x2 =8 (bounded by 8 cubes)



Square: Imagine that…”a Cube, moving in some altogether new direction, but strictly according to Analogy, so as to make every particle of his interior pass through a new kind of Space, with a wake of its own - shall create a still more perfect perfection than himself, with sixteen terminal Extrasolid angles, and Eight solid Cubes for his Perimeter. And once there, shall we stay our upward course? In that blessed region of Four Dimensions, shall we linger on the threshold of the Fifth, and not enter therein? Ah, no! Let us rather resolve that our ambition shall soar with our corporal ascent. Then, yielding to our intellectual onset, the gates of the Sixth Dimension shall fly open; after that a Seventh, and then an Eighth -…”

Of course, Sphere couldn’t comprehend such possibilities!

The mathematical reasoning exhibited here by Square is common in logic.  It is a process of extrapolating from a pattern.  Such pattern questions are often used in IQ tests.  The subject is asked to state the simplest logical number that would come next in a given series of numbers.  Here are a couple of examples:

What comes next?
    (a) 3, 4, 8, 9, 18, 19, ___
    (b) 1, 1, 2, 3, 5, 8, 13, ___
In (a) the sequence consists of a pair of consecutive numbers, followed by a number which is twice the size of the last number.  Continue: consecutive, double, consecutive, double… The next logical number after 19 would be 38.

In (b) the sequence starts with 1, 1.  The third number is formed by adding the previous two.  2 = 1+1.  Then 3=2+1, 5=3+2, 8=5+3, and 13=8+5.  The next number would be 21.  By the way, this special sequence is called the Fibonacci Sequence – and it will be the subject of an upcoming article.


The Triangle "Experiment"

Mathematical reasoning, Geometry, and many topics in Physics are closely connected to the topics of space and dimensions. Albert Einstein’s Special Theory of Relativity uses time as the fourth dimension and describes our universe in terms of a 4-dimensional space-time system. In addition, Einstein tried to produce a unified theory that would explain all of the different forces in the universe, including gravitational, electromagnetic, and atomic forces in his General Theory of Relativity. He wasn’t entirely successful in the latter quest, but his initial work has been carried forward by scientists and mathematicians ever since. Superstring theory has had some success, but its theory requires that we exist in a 10-dimensional (or is it 11?) space-time system! Try to wrap your mind around that concept! (Interested? Watch the video “String Theory: Welcome to the 11th Dimension” in the “Resources” section at the end of this article).

Mathematicians often come up with results in theory that scientists have to try to explain with realities. Let me invent a scenario that I will relate to the world of Flatland.

Let’s assume that Flatland actually exists on the surface of a sphere – similar to the way we live on the planet earth. (Many centuries ago, some people believed that the earth was flat, and that if we sailed far enough out onto the oceans we would fall off the edge of the world. Today, some people still believe this to be true i.e. Flat Earth Society). If Square and his cohorts had lived on the surface of a sphere, geometry may have led them to a surprising realization.

In last month’s article, “Euclidean Geometry”, we saw that one of its postulates – or assumptions – was that only one line can be drawn through a point that will be parallel to another given line (the Fifth Postulate). One of the theories, or proofs, that result from this assumption was that the sum of the interior angles of any triangle is 180° (degrees). This fact is correct only if the triangle lies on a flat surface (known as a plane surface). It is not true for triangles that lie on curved surfaces.

non-042The inset of this picture shows a blow-up of a triangle drawn on a tiny portion of the earth’s surface, with three angles whose sum appears to be 180°, (as predicted by Euclidean Geometry). Another large triangle is drawn with its base sitting on the equator of the earth; its other two sides run vertically along lines of longitude and meet at the North Pole at an angle of 50°. The sum of the angles in this gigantic triangle is 230°. If different lines of longitude were used, the angle at the pole could have been smaller or larger, so the sum would have been different from 230° – but the sum would have been larger than 180° in all cases.

It is difficult, if not impossible, for us to construct such a large triangle on the actual surface of the earth because of undulating terrain, but it could be done from the peaks of three distant mountains, using lasers for example, and the angle measurements could be made using accurate technological tools. It would be much easier, however, for the inhabitants of Flatland because there would be no obstructions on the smooth flat surface of their world. Suppose Square had been able to draw a series of larger and larger triangles and measure their angles. What would he think when the mathematical measurements showed the sum of the interior angles of a triangle was something other than 180°? 181°? Or 182°? 185°? 230°? He would ask what it meant. What reality is being described by the mathematical results? Our scientists are trying to do the same thing when they study the data being collected from investigations into the nature of our universe.

Spherical Geometry

There are many different, logical and consistent geometries. Euclidean Geometry, also known as Plane Geometry, applies to shapes drawn on flat surfaces. Remember, the origins of geometry spring from practical applications such as land measurements. Recall some of the facts that are associated with it:

  • A triangle drawn with three straight lines that intersect at three points (called vertices), always has 180° as the sum of its three interior angles.
  • Through any given point P that is not on a given line, only line can be drawn that is parallel to the given line. In essence, this means that two parallel lines never meet – and are always exactly the same distant apart from each other.

non-043On a sphere, things are different. As we have seen, straight lines are actually curves that follow the rounded surface. Only the “great circles” (or geodesics - the lines of circumference) are defined as the ‘lines’ in this geometry. Triangles that are drawn with such straight lines have interior angles that add up to more than 180°. On earth, lines of longitude cross the equator at 90° angles and they seem to be parallel to each other at that point, but they do not remain the same distance apart as they move further away from the equator - they get closer and closer together and eventually meet at two points, on opposite ends of the earth (see diagram). In this geometry, called Spherical Geometry, any two straight lines cross at two points: there are no parallel lines. Spherical Geometry is a subset of a broader form of geometry called Elliptical Geometry.

While most of us can function throughout our lives by using only Euclidean Geometry for our measurements, some professions, like aircraft navigation, rely entirely on Spherical Geometry.

Hyperbolic Geometry

non-044The surface for this type of non-Euclidean geometry is usually depicted as “saddle-shaped”, but there are other surfaces that display the same characteristics. Other models have been produced by crocheting (these models very closely resemble the structures of coral and some sea slugs; examples are shown at the end of this section). The illustration at the left shows a triangle on the surface of a saddle. The sum of the interior angles of any triangle on such a surface is always less than 180°.

non-047The straight lines on these surfaces curve like the straight lines on the surface of a sphere – but ‘outwards’. Diagrams like the one at right illustrate another type of triangle on this type of curvature.

non-048The “Parallel Postulate” from Euclidean Geometry would produce an entirely different result in Hyperbolic Geometry. Through any given point ‘P’ that is not on a given line, an infinite number of lines can be drawn that are ‘parallel’ to the given line (that is, they never meet it) – as illustrated in the diagram at the right.

Coral formations, some sea slugs, some types of leaf lettuce, the tentacles of jellyfish, and various crochet patterns display hyperbolic geometry structures. This form of geometry is necessary in the study of cosmology (in astronomy). It is also used in the study of the properties of complex networks, in crystallography, and in modelling the human vision system.


Applications of Various Geometries


History of Non-Euclidean Geometry

Euclid was a Greek mathematician who lived approximately 2,300 years ago. His book “Elements” documented the logical approach to the study of plane geometry and served as the basis for the formal study of geometry for over 2,000 years.

In the nineteenth century a number of mathematicians questioned Euclid’s Parallel Postulate (the Fifth Postulate), and their work led them into new directions and ways of thinking about geometry. Instead of using Euclid’s assumption that only one line could be drawn through a given point that would be parallel to another given line, they studied the implications of replacing this postulate with one of its negations:

  • through a point not on a given line, there is no line that can be drawn parallel to the given line
  • through a point not on a given line, there is more than one line that can be drawn parallel to the given line

Janos (or Johann) Bolyai was one of the mathematicians who considered the possibility of an alternative to Euclidean Geometry. After some initial work, he could see that his thoughts were taking him in the right direction: there could be another logical, consistent geometry. In his excitement, he wrote to his father (who was also a mathematician), “I have created a new world out of nothing!” The ‘new world’ was Non-Euclidean Geometry. While Bolyai conceived his ideas in Hungary, another mathematician, Nikolai Lobachevsky, was independently developing the same conclusions in Russia.


Some of the key mathematicians involved in the development of non-Euclidean geometries are shown above. The following link will take you to an article that provides information about their lives and their contributions: History of Non-Euclidean Geometries. The Resources section also provides a link to a video on the same topic, but with a more detailed mathematical discussion.

The Curvature of Space

Square, living in two-dimensional Flatland, was astounded to learn there was a three-dimensional universe that included his world within it. Humans, since Einstein, have questioned whether our three-dimensional world exists inside a multi-dimensional universe. Some theories indicate that we do, while others go beyond that and suggest that we are part of a multiverse (i.e. that there are many parallel universes in which we exist). It is a strange and complex picture!

non-062Over 100 years ago, Einstein (1879 – 1955) began to question many things, including the laws of motion, the concept of time and the nature of gravity. His work on gravity challenged the notions of Newton (1643 – 1727) that gravity was a force of attraction that acted instantly at a distance. Instead, he speculated that the fabric of space could be distorted by the presence of bodies – in much the same way that the surface of a trampoline is distorted by a human who jumps onto it. More massive objects, such as stars, create deep ‘gravity wells’ which affect other bodies that come near. non-062b A ball will tend to roll in a straight line on a flat surface (as in diagram (a)), but its path will be deflected, or bent, as it approaches the region near such a well (diagram (b)). It might even become trapped, and roll around in circles or ellipses around the sides of such wells (like orbiting planets). Einstein predicted that light itself would be bent by massive objects near its path. This prediction was verified many years later by observations of the positions of stars near the edge of the sun taken during a solar eclipse, compared to observations of those same stars at a time when the sun was not near their line of sight with the earth.

non-064Space is not just “empty” – but has characteristics that enable it to be distorted, bent, stretched and rippled – like the surface of a quiet pond of water. Different locations in space would obey different laws of geometry, depending on the curvature of space in those locations.

Our universe is a strange and wondrous place. Three-dimensional human beings are trying to decipher the mysteries of our universe using all of the tools available to us. We are like Square, living in Flatland, trying to imagine a universe beyond what our eyes perceive. The tools for our imaginations are Logic, Physics and, of course, Mathematics.—

This article is the second of a series of mathematics articles published by CHASA.

Marvellous Mathematics – Introduction

# 1 Euclidean Geometry - Issue 23 - April 1, 2014

# 3 Rational Numbers – Fractions, Decimals and Calculators - Issue 25 - June 1, 2014


CHASA has received many communications from concerned parents about the difficulties their children are having with the math curriculum in their schools as well as their own frustration in trying to understand the concepts - so that they can help their children.

The intent of these articles is to not only help explain specific areas of history, concepts and topics in mathematics, but to also show the beauty and majesty of the subject.

Dave Didur is a retired secondary school mathematics teacher with a B. Sc. degree from the University of Toronto majoring in Mathematics and Physics. He was Head of Mathematics for over twenty years, as well as the Computer Co-ordinator and consultant for the Board of Education for the City of Hamilton. He served with the Ontario Ministry of Education for three years as an Education Officer.



Flatland: A Romance of Many Dimensions – Abbott’s book in its entirety, with drawings


Extra Dimensions Explained (Video) – a clip from the movie “What the Bleep!? Down the Rabbit Hole” that shows how a Flatlander learns about an ‘extra’ dimension

non-066Hypercubes – interesting relationships of n-cubes

Looking for Extra Dimensions – part of the official String Theory website

Flat Earth Society – what is it?

Top 10 Reasons ‘Why We Know the Earth is Round’ Debunked – a Flat Earth Society forum in which the “Triangle Experiment” is questioned – sort of…

Non-Euclidean Geometries:

Euclidean and Non-Euclidean Geometries – a nice overview

Non-Euclidean Geometries – Euclidean, Spherical, and Hyperbolic Geometries are discussed and well illustrated

Spherical Geometry – Wikipedia notes

Truth, Beauty, Math and Crocheting – Hyperbolic Geometry in the structure of corals and crochet works

Margaret Wertheim’s TED Lecture (Video) – About crocheting Hyperbolic Geometry structures

Geometries are Empirical – University of Pittsburgh lesson from the “History and Philosophy of Science” course

Geometry Junkyard – links to many advanced topics related to Hyperbolic Geometry

History of non-Euclidean Geometry:

History of Non-Euclidean Geometries – some of the key mathematicians

Non-Euclidean Geometry History (Video) – a more detailed mathematical look at the topic, questioning Euclid’s Parallel Postulate

The Curvature of Space:

Gravity As Curved Space: Einstein’s General Theory of Relativity – basic notes

Einstein’s Theory of General Relativity – advanced notes, with experiments that have been conducted to prove Einstein’s predictions

Introduction to General Relativity – a broad discussion of topics

Space-Time Vortex Around Earth (Video) – 2004-2011 NASA Experiment testing part of Einstein’s theory

Gravity Probe B: Testing Einstein’s Universe – an excellent web site that gives good detail about Einstein’s works and the mission to test his thoughts about gravity (the mission is highlighted in the video “Space-time Vortex Around Earth”)

What is the Geometry of the Universe? – a layman’s discussion of the fate of the universe


Cosmology – scientific discussion of the evolving universe

The Curvature of Space (Video) – Gravitational effects illustrated as the result of curved space

Measuring the Curvature of Space – using very long baseline array (VLBA) radio telescopes

Parallel Universes (Multiverses) – how many of you are out there?

The Extra Spatial Dimensions of String Theory (Video) – tiny curled up extra dimensions in our universe


Read 34276 times Last modified on Thursday, 03 December 2015 11:12



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