triangle Index

My new Conics pages

I have spent much time over the past few year working on conics. I have found new ways to classify conics in the triangle plane. The conics hub contains links to hundreds of files.

John, Adam, and Steve's excellent adventure: The Hexagrammum Mysticum

The John here is John Conway, who is already famous. The Adam is Adam Marcus, who will be famous. The Steve is me, who will never be famous. The adventure is our research into Pascal's Hexagramum Mysticum. Fun, fun, fun. But then, Geometry is fun.

6/20/06 –updated with more information about groups and the color symmetry. 8/14/06 – updated again, this time with more group information and information about Pascal hexagons.

New Book: King of Infinite Space; Donald Coxeter, the man who saved Geometry
by Siobhan Roberts
A forthcoming book about the great geometer of our time. It is available now.

Geometry Quote from Casey's Modern Sequel to Euclid:
Observation.–...a convention universally adopted by modern geometers, namely–when a geometrical magnitude of any kind which varies continuously according to any law, passes through the zero value to give it the algebraic signs, plus and minus, on different sides of the zero–in other words, to suppose it to changes sign when passing through zero.

Triangle geometry

Euclid invented a geometry of points, segments, and circles. Constructions used ruler and compass. Our geometry is of points, lines, and more complex curves. Our constructions are more varied and often done using a computer.

Our time is one of discovery, the third great age of triangle geometry! A problem is the sheer number of discoveries. Particle physics, my original interest, went through a similar period. The basic particles of proton, neutron, and electron nicely explained atoms and properties of materials. But it was followed by an explosion of unexpected and unexplained new particles. "The mu-meson," Pauli is alleged to have said at dinner, "who ordered that?" Similarly we geometers might say, "the Clawson cotangent point; who ordered that?" Just as the particle physicists found the answer in symmetry, so too do we. This is math; answers are always found in symmetry! And as in physics the quantity of discoveries becomes a research topic of its own.

Simple notation: As our subject matures, notation should also mature. Simple things should be expressed simply; the number of operations minimized and jargon understandable to only a few eliminated.

A bigger beast: When a geometric structure can be described with unexpected simplicity, we trace this simplicity back to the great groups of transformations known as projective, affine, inversive. But they too have their rules, their symmetries. Mathematicians been busy in the last hundred years; there are bigger beasts in the geometry jungle out there!

Wider (and wilder!) structure. With so many new discoveries, new and wilder structure may be found. This is what I am hoping to find. In fact it is what I think I have found.

The distribution of significant points in the plane of the triangle that interests me more than the individual points themselves. Check out the Orbits and sweeps hub and the affine theory of the triangle, my attempt (I believe successful) to give a place to every point, line, and conic.

The conics project: Information about triangle conics is not as good as triangle points and cubics, so I am making my own catalog of conics. See the Strong Conics Hub and the Weak Conics hub. These are interactive. Click on the name of the conic you wish to explore. I have found out much about the points on conics for which the best explanation and examples are here.

I have been applying purely algebraic techniques which have been amazingly fruitful. The foundation for this approach is explained in "From Euclid to Abstract Algebra." Its culmination is found in the articles "The Non-Euclidean Geometry of Triangle Centers," "Natural Points, Lines, and Conics ", Points on GP, and "Distribution of Centers and SuperStrong Points."

Recently revised : Most pages dealing with conics are being revised as I learn more about them : Conic Types , most conic pages.

Recently Added: Strong Conics Hub and the Weak Conics hub, The Steiner ellipse ; axes and asymptotes for circumconics; Mandart inconics,Triangle Conics, Inversion Fractals ; a Mathematica computation of the asymptotes of a circumconic. A very interesting picture of isotomic directions and parabolas inscribed to the medial triangle is posted.

Posted but under revision : Infinity and Below–Points on Conics , Points on Steiner Ellipse and on Circumcircle

Note: Many of the hyperbola documents contain links to Sketchpad documents which contain tools for the relevant ETC points in all versions.

The pictures on this site (there are thousands) were created in Geometer's Sketchpad or Mathematica and usually finished in Adobe Illustrator. The above picture is of Lemoine's antiparallel circle and was created by John Conway and myself.

Status of The Triangle Book

For those who do not know I am coauthor with John Conway.of the forthcoming The Triangle Book. The goal of the book is to definitively describe triangle geometry. The current plan is that it will be shaped like a triangle, full color every page, and the height of an atlas, although financial considerations will make the final call on this. A.K. Peters is to be the publisher. (Here is a sample page). Most of the text is written but progress is slow in finishing it because we are only together about 3 weeks a year. John works on a number of books at once, each with a different coauthor or set of coauthors. When I as coauthor walk in the room I get his undivided attention, which is an amazing thing to get. But as soon as I walk out of the room, John is off to something else. More math to be created! Each time we get together, we add as much new material as we finish old material. For example, at my suggestion we wrote a section on the Pascal Hexagrammum Mysticum. It was so much fun, and so fruitful, that we spent a week just on this. New mathematics was created, which is always more fun than finishing a book, and a new section was added to the book. The Princeton mathematicians keep telling us that "a book is never finished; it is merely abandoned." We have not yet abandoned it. I figure it is still a year away from completion, partly because John will be traveling a lot this year so that it will be harder for us to get together (unless I can go to New Zealand ... hint...hint!).

In meantime, some of the pictures and a little bit of triangle book content can be found here.

Triangle book notation can be found here at the bottom of this web page.

So you want to learn triangle geometry

Triangle geometry is a combination of the very old and very new. If you want to get quickly into the subject, here is lesson 1 on the projective geometry of the triangle. This lesson is easy to execute and quickly gets you to the most important topics. Other lessons follow. By hand or by computer draw lots of pictures.

Lesson 0: learn about the 4 classical centers, the incenter, the centroid, the circumcenter, and the orthocenter. Use the web for this.
Lesson 1: the projective geometry of the triangle.
Lesson 2: the Euler line is a piece of cake.
Lesson 3: The symmedian, Gergonne points, and Nagel points. Link to Paul Yiu's excellent introduction.
Lesson 4: The incenter comes in four versions.
Lesson 5: The incenter and the circumcenter are different.
Lesson 6: Antiparallels and Lemoine Geometry, the discovery that changed triangle geometry.
Lesson 7: Deviation from centeredness; conjugates
Lesson 8: The line at infinity and the Steiner ellipse, the affine theory of the triangle
Lesson 9: The inversive geometry of the triangle

What's here in these pages

I love triangle geometry so this is an enthusiast's webpage. I put in things that I find interesting. I like making pictures, so there are lots of those. There are pdf articles on aspects of triangle geometry. If you wish to learn the new ways of doing triangle geometry, try the page on "Trilinear lines" below and check out the other pdf's. Links titled"hub" are actually host pages for a number of pages on the stated topic.

To the left, Top: links to my (usually current) work in varying states of exposition.
Left, Middle: Pictures and short expositions, some from the triangle book; others that I have generated.
Left, Bottom: Conversations with Conway.
Below: Links and descriptions of previous geometric interests. As things get finished they will move from the current works (above left) to a link and description below.

Desargues This page on Desargues configuration is interesting as is the Desargues slideshow. The important thing here is the notation that makes it obvious that S(5) is its symmetry group. The Desargues' configuration has 10 points and 10 lines incident in three's.	Geometry tutorials (pdf's) When I was learning triangle geometry about 10 years ago, I wrote articles for myself and for friends. These articles are all in pdf format. I recommend beginning with trilinear lines, described in the left column. Conics -- a description of conics using homogeneous coordinates and the method of line (see Trilinear lines). I learned this from English books from the 1880's. Modern Geometry of the Triangle -- a description of basic things a person should know about the more recent (1800's) developments in geometry. Pictures -- I like making pictures; here are some I did about 8 years ago.
The case of the missing tangents The Case of the missing tangents. A slide show in the modern geometry of the triangle.	Geometrical cubics Cubic curves are as powerful an organizing mechanism as we have in triangle geometry. They have a group table whose foundation lies outside the subject and represents a powerful addition to our subject. A good way to begin this subject is to read Mineur's Anallagmatic cubics linked to the left. These pictures show both the cubic (some in my new style of drawing cubics) and its group table.
Mineur's Anallagmatic cubics An anallagmatic structure includes it opposite. In triangle geometry the opposite is called the conjugate. These pages transcribe a 1922 monograph by Adolph Mineur. As geometry goes this is pretty advanced, using barycentric coordinates and general knowledge of triangle geometry, especially projective triangle geometry. I translated this from the French original which I came across in the Princeton Mathematics library.	Trilinear lines (pdf) This is a new way of thinking about lines. I wrote this about 10 years ago when I was beginning to work on Triangle geometry. I had purchased a second hand book on Conics by Issac Todhunter (one of my favorite old textbook writers). I looked at the early chapters just to get used to the notation. The first two chapters were about straight lines. To my surprise there was a lot that I did not know about straight lines ( I thought I know everything about them -- after all they are straight, they have a slope and intercepts, there are vector forms -- what else was there?). Well, straight lines can be written using homogeneous coordinates; this was new to me so I wrote it up, both to understand it myself and to pass it on. This is pdf file rather than a web page. If you wish to learn the modern way of doing triangle geometry, this is a good place to begin.
Barycentric coordinates This is a method of letting a two dimensional point in the plane of the triangle be represented by three coordinates. Rather than have an origin for the coordinate system, we use a reference triangle instead. This type of coordinate system is useful when the numerical value of distances is not important, and objects are not dependent on scale. There are two parts. One show how to find Barycentric coordinates. Finally there is a long email message from John Conway on the superiority of the barycentric system.	Abstract Algebra and Triangle Geometry We are in the third great period of discovery in triangle geometry. The first was the development of the synthetic method by the Greeks; the second was the transition to algebra and a wider view of geometry done in the 19th century. Now we have computers and abstract algebra. The computers tie us all together and allow us to compute quickly and draw quickly. By tying us all together, many points of view are quickly mixed and vetted, to the benefit of all. But mathematics has moved way past triangle geometry, and the methods developed by them are available to us now. The extraversion symmetry is a new addition to geometry that comes from Galois theory (Kapetis' version) or monodromy theory (John's version). Much algebraic geometry is a development of the properties of cubic curves, and of direct relevance to our subject. An excellent example of the use of abstract considerations is the liked work above on the incentral triangle and excentral triangle.

Steve Sigur's Webpages -- Triangle Geometry

Triangle Geometry

Pictures from The Triangle Book and, increasingly, my stuff

Conversations with John Conway

My new Conics pages

John, Adam, and Steve's excellent adventure: The Hexagrammum Mysticum

Triangle geometry

Status of The Triangle Book