https://pi.math.cornell.edu/~mec/mircea.html
Non-Euclidean Geometry Online: a Guide to Resources
by
Mircea Pitici
June 2008
Good expository introductions to non-Euclidean geometry in book form
are easy to obtain, with a fairly small investment.[1] The aim of
this text is to offer a pleasant guide through the many online
resources on non-Euclidean geometry (and a bit more). There are also
three instructional modules inserted as PDF files; they can be used
in the classroom.
The geometry around us
Geometry must be as old as humans' struggle for survival. Building a
good hunting bow and getting the best arrows for it surely involved
some intuitive appreciation of space, direction, distance, and
kinematics. Similarly, delimitating enclosures, building shelters,
and accommodating small hierarchical or egalitarian communities must
have presupposed an appreciation for the notions of center,
equidistance, length, area, volume, straightness.
Some of these deceptively "clear" terms remain more ambiguous than a
cursory view accords them. We are not always well served by the
millennia-long mathematical acculturation that pervades even our best
available instruction in school geometry.
What is geometry?
Curious geometrical patterns are ubiquitous. Next time you are in the
produce section of a supermarket, take a close look at some fruits
and vegetables with particularly interesting configurations. Look,
for instance, at the Romanesco broccoli, at kale and lettuce , and at
the pineapple. Or walk over to the floral department and search for
flowers with sophisticated conformations. You might see arrangements
no less interesting than the ones discernable in the vegetables. Here
are some computerized renderings.
The beauty embedded in these naturally occurring patterns is not only
pleasurable but also intriguing. Over the last few decades a growing
number of mathematicians worked on making theoretical advances in the
study of patterns similar to the ones pictured above. They found
suggestive models useful in understanding such patterns and
discovered new applications.
Two mathematical fields are particularly apt for describing such
occurrences: the theory of fractals and non-Euclidean geometry,
(especially hyperbolic geometry).
In this text we will limit ourselves to the latter. The organization
of this visual tour through non-Euclidean geometry takes us from its
aesthetical manifestations to the simple geometrical properties which
distinguish it from the Euclidean geometry.
There are two main types of non-Euclidean geometries, spherical (or
elliptical) and hyperbolic. They can be viewed either as opposite or
complimentary, depending on the aspect we consider. I will point out
some of the theoretical aspects in the final sections of these
presentation.
Hyperbolic geometry and handcrafts
Since 1997, when Daina Taimina crocheted the first model of a
hyperbolic plane, the interest in "hyperbolic handcrafts" has
exploded. The imagination of the crafters is unbound. The reader can
explore a wealth of such artifacts on the website of the Institute
for Figuring founded by Margaret Wertheim.
With the proliferation of hyperbolic crocheting, more and more
exhibitions hosting this work open around the world. Currently
(Summer 2008) one is open at the Hayward Southbank Centre Car Park in
London, under the title Hyperbolic Crochet Coral Reef, part of the
Hyperbolic Coral Reef Project of the Institute for Figuring mentioned
above. Other examples of hyperbolic crocheting can be found online
here, here, and here.
An excellent starting point for people interested in learning more
about this subject is Sarah-Marie Belcasto's mathematical knitting
pages. The website has many other internet links, including a brief
bibliography of written resources of a similar nature, and other
photos of hyperbolic knitting. Finally, a few more hyperbolic
crocheted flowers can be viewed on this site.
The subject was also covered by the national press in the US, for
instance in this recent New York Times article, this older one, in
Britain's The Guardian, as well as on the National Public Radio.
At Cornell, Daina Taimina's crocheted models are essential tools for
teaching hands-on and for understanding hyperbolic geometry in
high-level undergraduate courses. Her manipulatives are highly
versatile. Students experiment with geometrical objects otherwise
difficult to visualize and test their intuition of geometrical
properties. In a 2004 interview, Cornell's Daina Taimina and David
Henderson explain the historical background and the differences
between the geometrical properties of the planar, spherical, and
hyperbolic geometries.
The well-known writer on mathematics Ivars Peterson published a
column in 2003 on hyperbolic geometry in art and crafts. He
exemplifies the building of an "unruly quilt" from cloth-made
pentagons sewn together. This construction is well known among the
mathematicians preoccupied with hyperbolic geometry. Its theoretical
importance (and that of one variation) are discussed in the following
instructional module, inserted here in PDF file:
Non-Euclidean
A Google search for the "water strider" mentioned in this module
produces numerous descriptions and images. Apparently, the insect
even inspires technological advances in robotics.
Art and hyperbolic geometry
The visual potency of the hyperbolic models of non-Euclidean geometry
has captured the imagination of artists.[2] The most famous painter
to have used hyperbolic models was M.C. Escher. His work is now
widely known. The relationship and historical avatars between the
circular model of hyperbolic geometry and Escher's art are presented
in this article by Douglas Dunham of the University of Minnesota at
Duluth, who also wrote this article on the same subject. Many more
resources on Escher's art and mathematics can be found on this
website.
Recent echoes of non-Euclidean shapes found their way in architecture
and design applications.
Academic resources for the study of hyperbolic geometry
We now turn our attention to online resources on non-Euclidean
geometry (mostly hyperbolic geometry) made available by educational
institutions or clearly directed toward instructional purposes.
Geometry Technologies, whose goal is "to bridge science and math to
life through software," hosts and excellent Geometry Center online,
with a good section on hyperbolic geometry.
More visualizations of curved spaces are available online on a
website maintained by Jeffrey Weeks, the author of the book The shape
of space, mentioned in the first endnote to this text. Under the
title 'Curved spaces' it contains applets able to take you on a
curved spaceship through 3-Torus, a hyperbolical dodecahedron, a
Poincare dodecahedral space, and a spherical cube. A shortened
version of a 1994 article published in the magazine Mathematical
intelligencer also has visualizations of several pseudospheres.
A combination of fractal art and hyperbolic geometry can be found at
the Hidden Dimension Galleries.
A richly illustrated chapter in all types of geometries is provided
by tiling, the problem of covering a given surfaces with shapes of a
given form, without gaps or overlaps. A versatile applet for
hyperbolic plane tiling with triangles lets the experimenter play
with various parameters. Here is another applet, for covering the
circle in the Poincare model.
Other good online resources on non-Euclidean geometry can be found on
the Visual Mathematics website; the Geometry Junkyard, with its
Tilings of Hyperbolic Spaces page; as well as on the pages maintained
by independent groups of enthusiasts, like this, or this. The
Wikipedia page on non-Euclidean geometry also has merits. Much weaker
in terms of theory (but good for some bibliographical references) is
the entry on non-Euclidean geometry in Wolfram MathWorld.
Several websites offer excellent dynamic software. One is Topology
and Geometry Software maintained by Jeff Weeks. Another one is
Geometry and motion, maintained by Daniel Scher.
Drexel University maintains a remarkable Math Forum online, with a
good page on hyperbolic geometry. Another one is maintained by a
group of three mathematicians at various universities.
The basics
For millennia, the idea that no-Euclidean geometries might exist was
anathema among mathematicians. Karl Friedrich Gauss, arguably the
best mathematician ever, delayed publishing his research on
non-Euclidean geometry, fearing that he could compromise his
reputation.
Throughout the last two centuries several intuitive models of
non-Euclidean geometries were proposed. In most of them the
definitions of basic geometrical notions challenge our commonly held
spatial intuitions. They are, nonetheless, self-consistent within the
model to which they belong.
So what are non-Euclidean geometries?
Many authors (including some mentioned in this text) wrongly assume,
as a matter of definition, that logically self-consistent geometrical
structures that do not abide by Euclid's fifth postulate, are
non-Euclidean. Yet spherical geometry - which is non-Euclidean - does
abide by Euclid's fifth postulate. The confusion stems from the fact
that Euclid's fifth postulate and its logical equivalents in planar
geometry are not always equivalent in non-planar geometries. The best
discussion of this topic can be found in Henderson & Taimina's book
quoted in the endnote 1. In chapter 10 the reader finds a detailed
table comparing various forms of Euclid's fifth postulate in planar,
hyperbolic and spherical geometries.
One of the many ways of comparing these geometries and the planar
Euclidean geometry is to look at the sum of the interior angles of a
triangle in each of them. In the spherical geometry the interior
angles always add up to more than two right angles (180 degrees), in
the planar geometry they add up to exactly two right angles, while in
the hyperbolic geometry they add up to less than two right angles.
Here is an example of a triangle on a sphere, with three right angles
(adding up, therefore, to 270 degrees):[3]
[tripleright]
and another one, in which all angles exceed a right angles and the
triangle's area (the shadowed part) is almost as big as the whole
spherical surface:[4]
[largesphericaltriangle]
From Daina Taimina's collection of crocheted hyperbolic planes, the
following has a triangle marked in light colors; the measures of the
angles add up to less than 180 degrees:[5]
[hyperbolictriangle]
It can be shown that in each type of non-Euclidean geometry the sum
of the interior angles of a triangle is directly related to the area
of the triangle. Also, the area of a geometrical figure depends on
"how much curved" the surface is - on the curvature of the surface.
For a spherical surface we can loosely speak about an 'area deficit'
due to its curvature (since we get area "gaps" if we flatten it onto
a Euclidean plan):[6]
[flattening]
To the contrary, in the case of a hyperbolic surface we can speak
about an "area surplus" or abundance (since we get overlaps if we
flatten it onto a Euclidean plan):
[flattening-a-hyperbolic-plane]
Area of a circular surface grows differently in each type of
geometry. In Euclidean planar geometry it grows proportional with the
square of the radius of the circle. In hyperbolic geometry it grows
exponentially with the growth of the radius. In spherical geometry
the area grows with the radius but it cannot exceed the area of the
whole spherical surface.
The three geometries also differ is the system of coordinates best
implemented in each. This issue is of great importance for the
computational treatment of each type of geometry. We are widely
acquainted with the rectangular system of coordinates for the
Euclidean plane.
For a sphere, the most familiar coordinate system is the latitude/
longitude grid used in geodesy to identify locations of the surface
of the earth. Yet that one is not without ambiguity, as shown in a
public radio interview on the subject of Hurricane Katrina. I insert
here an instructional module focused on the two most commonly used
coordinate systems, planar and spherical:
Coordinate systems
On a hyperbolic plane the most convenient system of coordinates is
also rectangular, as shown in the following picture:[7]
[hyperboliccoordinates]
More technical treatment
Non-Euclidean geometry can also be introduced and studied in a highly
technical manner. For the reader interested in such an approach we
offer a brief bibliography.[8] As one would expect, the online
resources are more limited, but not inexistent.
--------------------------------------------
[1] Even without a strong mathematical background, interested readers
can find excellent introductory books into the problematic of
non-Euclidean geometry. A few I would recommend are the following (in
alphabetical order of the authors):
* Bolyai, Janos. Non-Euclidean Geometry and the Nature of Space.
Ed., with a (long) commentary, by Jeremy J. Gray. Cambridge, MA:
MIT Press, 2004. A detailed historical account introduces the
reader to the battle of ideas around non-Euclidean geometries.
* Henderson, David W., and Daina Taimina. Experiencing Geometry:
Euclidean and Non-Euclidean, with History. Third edition. Upper
Saddle River, NJ: Prentice Hall, 2005. This is a textbook used in
several undergraduate courses in the U.S. and Canada. It provides
an inviting, detailed, hands-on, inquiry-based approach to
learning non-Euclidean geometry. Especially instructive is the
comparative view, property by property. The bibliography,
outstanding even in the volume, is updated and annotated online
at http://www.math.cornell.edu/%7Edwh/biblio/ . Also good (some
groundbreaking) are the illustrations.
* Krause, Eugene F. Taxicab geometry: An adventure in non-Euclidean
geometry. New York, NY: Dover, 1975. This slim booklet is highly
entertaining. It contains many exercises in accessible format.
* Prekopa, Andras, and Emil Molnar (Eds). Non-Euclidean geometries.
New York, NY: Springer, 2006. This is a collective volume
published in the memory of Janos Bolyai. It contains
contributions of great variety, both in approach and difficulty.
* Trudeau, Richerd J. The non-Euclidean revolution. Boston, MA:
Birkhauser, 1987. Perhaps the most quoted book on non-Euclidean
geometry. The approach is more axiomatic than in other books.
* Weeks, Jeffrey R. The shape of space. Second edition. New York,
NY: Marcel Dekker, 2003. This is a highly readable introduction
to non-Euclidean geometries. Some of the figure captions toward
the end of the present web page are taken from Week's book.
[2] See Henderson, Linda Darlymple. The fourth dimension and
non-Euclidean geometry in modern art. Princeton, NY: Princeton
University Press, 1983. Most of the book is concerned with
multi-dimensional geometry.
[3] The image is taken from Henderson and Taimina's Experiencing
Geometry.
[4] Idem.
[5] Ibidem.
[6] The following two reproductions are taken from The shape of space
by Jeffrey Weeks.
[7] See Experiencing Geometry, chapter 5, for a detailed discussion.
[8] Several out of print books found in large academic libraries
offer good and succinct introductions in non-Euclidean geometry.
Obviously, they omit the visual/instructional developments that make
the object of this text:
* Buchanan, A. H. Plane and spherical trigonometry. New York, NY:
John Wiley, 1907.
* Kulczycki, Stefan. Non-Euclidean geometry. Oxford, UK: Pergamon
Press, 1961.
* Manning, Henry Parker. Non-Euclidean geometry. New York, NY:
Dover, 1901.
* Meschkowski, Herbert. Noneuclidean geometry. New York, NY:
Academic Press, 1964.
* Smogorzhevsky, A. S. Lobachevskian geometry. Moscow: Mir
Publishers, 1976.
* Somerville, D. M. Elements of non-Euclidean geometry. New York,
NY: Dover, 1958.
Among newer books, the following deserve attention:
* Greenberg, Martin Jay. Euclidean and non-Euclidean geometries:
Development and history. New Third edition. York, NY: W. H.
Freeman and Co, 1993.
* Rosenfeld, B. A. A history of non-Euclidean geometry: evolution
of the concept of a geometrical space. New York, NY: Springer,
1988. This book offers a thorough historical treatment, in
connection with other branches of mathematics.
More technical books include:
* Anderson, James W. Hyperbolic geometry. Second ed. New York, NY:
Springer, 2005.
* Canary, Richard, David Epstein, and Albert Marden. (Eds.)
Fundamentals of hyperbolic manifolds. New York, NY: Cambridge
University Press, 2006.
* Coxeter, H. S. M. Non-Euclidean geometry. 5th ed. Toronto,
Canada: U. of Toronto Pr., 1978
* Ramsay, Arlan, and Robert D. Richrmyer. Introduction to
hyperbolic geometry. New York, NY: Springer-Verlag, 1995.
* Stahl, Saul. A gateway to modern geometry: The Poincare
half-plane. Second edition. Sudbury, MA: Jones and Bartlett,
2008.
For a treatment that takes into account physics, see the following:
* Gray, Jeremy. Ideas of space: Euclidean, non-Euclidean, and
relativistic. Oxford, UK: Clarendon Press, 1989.
* Yaglom, I. M. A simple non-Euclidean geometry and its physical
basis: An elementary account of Galilean geometry and the
Galilean principle of relativity. New York, NY: Springer-Verlag,
1979.
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