School of Computer Science THE UNIVERSITY OF BIRMINGHAM CoSy project CogX project

A (Possibly) New Kind of (Non?) Euclidean Geometry
Based on an idea by Mary Pardoe.
(Hence the name "P-Geometry")

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NOTE: the "Non?" in the title was added August 2012 because this is a discussion of a
subset of Euclidean geometry extended with processes involving movable structures,
and therefore time.

It is not clear to me whether the result is a subset of Euclidean geometry or
something different.

NB: This is still work in progress, and likely to be changed/extended.
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Last updated:
21 Aug 2013
9 Dec 2010; 13 Dec 2010; 15 Dec 2010; 23 Dec 2010; 22 Jul 2011; 27 Jul 2012; 12 Aug 2012; 10 Sep 2012

Installed: 9 Dec 2010
Installed and maintained by Aaron Sloman
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Available in HTML and PDF

This file is http://www.cs.bham.ac.uk/research/projects/cogaff/misc/p-geometry.html
Also accessible as http://tinyurl.com/BhamCog/misc/p-geometry.html
From time to time I shall use html2ps and ps2pdf to create a PDF version, better suited for printing:
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/p-geometry.pdf
Or http://tinyurl.com/BhamCog/misc/p-geometry.pdf
The PDF version will probably have formatting flaws, and may not be up to date.
A partial index of discussion notes in this directory is in
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/AREADME.html

See also this discussion of "Toddler Theorems":
http://tinyurl.com/BhamCog/misc/toddler-theorems.html

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CONTENTS

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Introduction and overview

For over half a century I have been interested in the role of intuitive spatial
reasoning in mathematics. My Oxford DPhil Thesis (1962) was an attempt to defend
Kant's philosophy of mathematics, especially his claim that mathematical proofs
extend our knowledge (so the knowledge is "synthetic", not "analytic") and that
the discoveries are not empirical, or contingent, but are in an important sense
"a priori" (which does not imply "innate") and also necessarily true.

I had made my views clear in courses on philosophy of science and mathematics when
teaching at Sussex University (from 1964) which was why one of our former students,
Mary Pardoe (then Mary Ensor) who had become a mathematics teacher informed me, while
visiting the university, that she had found a new diagrammatic proof of the triangle
sum theorem. I reported her proof in some papers and presentations on methods of
representation and reasoning, e.g. here, but neither she nor I has encountered anyone
else who knew about the proof. So, in November 2010, I decided to try to get comments
on it from experts by writing to a mailing list on mathematical knowledge management (MKM).

The email discussions that followed helped to clarify what the new proof does and
does not assume (e.g. it does not seem to assume the Euclidean parallel lines
postulate). This web site is my attempt to summarise what I learnt from that
discussion. (Please see the acknowledgements section.)

In particular the discussion led me to attempt a partial formalisation of what I
think the proof assumes about moving, rotating, line segments as explained below.

There is a lot more work to be done, including clarifying the difference between
adding time and motion to Euclidean geometry and merely adding sets of locations (or
paths, loci) of geometrical entities to geometry without time.

NOTE: Updated 29 May 2013
I have produced two web pages that go into Pardoe's proof and other proofs of the Triangle
Sum Theorem in more detail, and also proofs of theorems about areas of triangles, and how
they change if parts move. These discuss some of biological and cognitive implications,
but without discussing how Euclidean geometry might be changed to accommodate processes
and time. See
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/triangle-sum.html
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/triangle-theorem.html
   (Mainly about processes that alter areas.)

Added: 22 Jul 2011
I have recently realised that my (still unfinished) attempt described below to produce a
version of Euclidean Geometry that does not include the parallel axiom but does allow
structure-preserving motion, can be construed as an example of what Annette
Karmiloff-Smith refers to as "Representational Redescription" in her 1992 book
Beyond Modularity: A Developmental Perspective on Cognitive Science
I have tried to explain this connection in my extended personal review
of her book, here:
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/beyond-modularity.html

Added: 15 Dec 2010

After producing a first draft of this web page I discovered that Henri Poincaré had
written an interesting discussion very closely related to the ideas here.

http://www-history.mcs.st-and.ac.uk/Extras/Poincare_non-Euclidean.html
Non-Euclidean geometries
by Henri Poincaré

Henri Poincaré published La science et l'hypothese in Paris in 1902.
An English translation entitled Science and hypothesis was published in 1905.
It contains a number of articles written by Poincaré over quite a number of years and we
present below a version of one of these articles, namely the one on Non-Euclidean geometries.

The following extract is directly relevant to the discussion below.

The Fourth Geometry.
Among these explicit axioms there is one which seems to me to deserve some attention,
because when we abandon it we can construct a fourth geometry as coherent as those of
Euclid, Lobachevsky, and Riemann. To prove that we can always draw a perpendicular at a
point A to a straight line AB, we consider a straight line AC movable about the point A,
and initially identical with the fixed straight line AB. We then can make it turn about
the point A until it lies in AB produced. Thus we assume two propositions - first, that
such a rotation is possible, and then that it may continue until the two lines lie the one
in the other produced. If the first point is conceded and the second rejected, we are led
to a series of theorems even stranger than those of Lobachevsky and Riemann, but equally
free from contradiction. I shall give only one of these theorems, and I shall not choose
the least remarkable of them. A real straight line may be perpendicular to itself.
________________________________________________________________________________

See also: Foundations of Geometry and Induction
By Jean Nicod
With an introduction by Bertrand Russell
Available on google books

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The sum of the angles of a triangle: The "standard" proof

There is a standard way (or small set of standard ways) of proving the theorem
Triangle Sum Theorem (TST):
The angles of a triangle add up to a straight line (180 degrees).

These standard methods all make use of some version of Euclid's parallel postulate,
which can be formulated in several equivalent ways, e.g.

Definition:
Two straight lines L1 and L2 are parallel if and only if they are co-planar and
have no point in common, no matter how far they are extended.

Postulate:
Given a straight line L in a plane, and a point P in the plane not on L, there
is exactly one line through P that is in the plane and parallel to L.

The "standard" ways of proving the TST make use of properties of angles formed
when a straight line joins or crosses a pair of parallel lines:

Corresponding angles are equal:
If two lines L1, L2 are parallel and a third line L3 is drawn from any point P1
on L1 to a point P2 on L2 and continued beyond P2,
Then the angle that L1 makes with the line L3 at point P1, and the angle L2
makes with the line L3 at point P2 (where the angles are on the same side of
both lines) are equal.

Alternate angles are equal:
If two lines L1, L2 are parallel and a third line L3 is drawn from any point
P1 on L1 to a point P2 on L2,
Then the angle L1 makes with the line L3 at point P1, and the angle L2 makes
with the line L3 at point P2 (on the opposite sides of both lines) are equal.

For more on transversals and relations between the angles they create see
http://www.mathsisfun.com/geometry/parallel-lines.html
The page teaches concepts with some interactive illustrations, but presents no proofs.

Two "standard" proofs of the triangle sum theorem using parallel lines are
illustrated here. as shown below:
two proofs

Warning: several online proofs seem to have bugs due to carelessness.
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Mary Pardoe's proof of the Triangle Sum Theorem

Many years ago at Sussex university I was visited by a former student Mary Pardoe
(then Mary Ensor), who had been teaching mathematics in schools. She mentioned that
her pupils had found the standard proof of the triangle sum theorem hard to take in
and remember, but that she had found an alternative proof, which was more memorable,
and easier for her pupils to understand.

Her proof just involves rotating a single directed line segment (or arrow, or pencil,
or ...) through each of the angles in turn at the corners of the triangle, which must
result in its ending up in its initial location pointing in the opposite direction,
without ever crossing over itself.

So the total rotation angle is equivalent to a straight line, or half rotation, i.e.
180 degrees, using the convention that a full rotation is 360 degrees. The proof is
illustrated below.
rotating segment

Available as printable PDF here.

An alternative presentation as a process
rotating
I have been presenting this proof in talks and papers on mathematical discovery and
reasoning for many years, until recently attributing it to Mary Ensor, as I had forgotten
her change of name. For instance in 
  Aaron Sloman, 2008,
  Kantian Philosophy of Mathematics and Young Robots,
  in Intelligent Computer Mathematics,
  Eds. Autexier, S., Campbell, J., Rubio, J., Sorge, V., Suzuki, M., and Wiedijk, F.,
  LLNCS no 5144, pp. 558-573, Springer,
  http://www.cs.bham.ac.uk/research/projects/cosy/papers#tr0802

  Aaron Sloman, 2010,
  If Learning Maths Requires a Teacher, Where did the First Teachers Come From?
  In Proceedings Symposium on Mathematical Practice and Cognition,
  AISB 2010 Convention, De Montfort University, Leicester
  http://www.cs.bham.ac.uk/research/projects/cogaff/10.html#1001

  And in talks on mathematical cognition and philosophy of mathematics here:
  http://www.cs.bham.ac.uk/research/projects/cogaff/talks/
The presentations produced no responses -- either critical or approving, except that in
an informal discussion recently a mathematician objected that the proof was unacceptable
because the surface of a sphere would provide a counter example. However, the surface of
a sphere provides no more and no less of a problem for Pardoe's proof than for the
standard proofs since both proofs are restricted to planar surfaces.

There may be interesting generalisations of both proofs that are applicable to
non-planar surfaces, but that's not the present topic.

I tried searching for online proofs to see if anyone else had discovered this proof
or used it, but nothing turned up. The proof using rotation is so simple and so
effective that both Mary Pardoe and I felt sure it must have been discovered previously.
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Email discussions of Pardoe's proof

In November 2010 I wrote to the Mathematical Knowledge Management Interest Group
(MKM-IG) mailing list, pointing at the online presentation of the proof and inviting comments.
I included a reference to Nelsen's "Proofs without words" (see below), as evidence that
there was nothing new in the idea that a serious mathematical proof can be based on
transformations of a diagram. In particular, I invited comments on the possible use
of the rotating segment proof in mathematics teaching.

From the ensuing discussion on the email list I was able to distill the following
alternative views. (Participants and others are invited to comment on this summary.)
  1. The proof is valid, but it is no different from the standard proof because the use of
    rotations adds rotations around different points in the plane and this requires use
    of the parallel postulate (since addition of angles that do not share a common vertex
    is otherwise undefined?)

    On this view the proof as presented is incomplete, and completing it would show it to
    be a variant of the standard proof.

    I responded to this that I thought there was an interpretation of the Pardoe proof
    that did not assume anything about the possibility of parallel lines or any
    properties of parallel lines, though it might be possible to deduce the parallel
    postulate from what it did assume, namely that a line segment can be rotated and that
    successive rotations can be summed even if they are rotations about different points.
    (Later I tried to draft a partial axiomatisation to show how the parallel axiom could
    be by-passed. See below.)

  2. A variant of comment (1) was the objection that the proof should not be used in
    teaching because as it stands it is not valid, and making it valid would require
    presenting axioms for Euclidean geometry that are assumed in the proof. This would
    then replace the Pardoe proof with something like the standard proof.

    I think this objection was based on an unproved assumption that it is impossible to
    prove the triangle sum theorem without assuming the parallel postulate (or something
    equivalent). The assumption is correct for some axiomatisations of Euclidean
    geometry.

    Since there have been different axiomatisations of Euclidean geometry over many
    centuries and rigorous formal axiomatisations were not possible before the
    developments of logic in the late 19th century, there must be a way of identifying
    what Euclidean geometry is that is independent of the way in which it is axiomatised
    formally.

    This seems to be connected with Kant's view that assuming something like Euclidean
    geometry is a requirement for perception and action involving objects in space. The
    refutation (by Einstein's work on relativity) of Euclid's 5th postulate as a
    characterisation of physical space still leaves a great deal of Euclidean geometry
    (including its topological subset and much more) intact.

    The assumptions made by the Pardoe proof may be part of that subset, especially if
    the proof can be shown NOT to assume the parallel postulate.
    (See the partial axiomatisation of P-geometry below.)

  3. Other comments pointed to various online demonstrations of mathematical properties of
    geometrical constructions, applications of pythagoras' theorem, algebraic formulae,
    etc. which could be used by students of mathematics to develop familiarity with the
    concepts and fluency in their use. Such development of mathematical intuitions is not
    the same thing as presenting a proof.

  4. In order to justify my assumption that the Pardoe proof was both intrinsically
    different from the standard proof, and to meet the objection, (1) above, that its
    assumptions are unclear, I decided to examine some existing axiomatisations of
    Euclidean geometry and thanks to google found Hilbert's and Tarski's axiomatisations,
    along with others.

  5. This led me to begin investigating the possibility of an alternative axiomatisation
    that included the notion of moving and rotating lines and other objects as primitive.
    I append the work in progress below, in the form of a partial axiomatisation.

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Formal vs Informal (including diagrammatic) mathematical proofs

There are some mathematicians and logicians who claim that only a formal (logical)
proof can be a proof (e.g. some followers of the Bourbaki school, whose influence I
believe seriously harmed the education of many young potential mathematicians).
In issue 73 of https://www.ncetm.org.uk/resources/28649 the NCETM secondary
magazine (edited by Mary Pardoe) there is a piece on mathematical education by Benoit
Mandelbrot, also relevant to this discussion.
To claim that only formal logical proofs are proofs ignores all the deep mathematical
advances over many hundreds of years (including Euclidean geometry) that preceded the
development of formal approaches.

Of course that begs the question: what makes a diagram, or a transformation of a
diagram (or anything else) a proof as opposed to a mere intuition-builder?

I think the full answer is very complex. It relates to the fact that when a structure
or configuration of any sort is perceived (or thought about) by humans and other
intelligent animals, though not yet by any robots I am aware of, that perceptual (or
thinking) state includes recognition and representation of

I recently discovered that the two last books Piaget wrote were on Knowledge of
Possibility and Knowledge of Necessity. I have not yet read them, but it seems that
he had noticed the points being discussed here.

J.J.Gibson's concept of perceiving and understanding affordances is a special case of
the concepts of knowledge of what is possible and what is necessarily the case in
various possible situations and processes. I don't believe Gibson, or most of his
followers understood the general point. A great deal of mathematical knowledge,
especially knowledge of geometry and topology, is concerned with facts about what
possible changes are possible in a given configuration and and what the constraints
on those possibilities and the consequences of those changes are (these are the
necessities).

The use of diagrammatic reasoning in mathematics depends on the fact that there are
ways of producing spatial structures and processes transforming such structures, that
reveal both possibilities for variation and also invariances at various levels of abstraction
that constrain those variations.

(Likewise logical or algebraic reasoning depends on the fact that there are ways of
transforming logical or algebraic expressions that reveal both possibilities for
variation and semantic invariances -- e.g. applying certain rules in a certain order
to a starting formula will necessarily produce a particular final formula (e.g. a theorem)).

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Metaphors vs Proofs

It became evident in some of the discussions and presentations at the Mathematical
Practice and Cognition Symposium at AISB2010 that some researchers on
mathematical cognition, especially non-mathematicians, seem to confuse diagrammatic
reasoning with use of metaphors. This is a serious error that is based on a failure
to understand the nature of mathematics and the difference between a suggestive
argument (which could be a metaphor) and a proof, which demonstrates what must
be the case.

I don't know to what extent the many people who are now building web sites for
mathematical education understand the difference. Both intuition builders and proofs
can transform the mind of the learner. But they do so in different ways -- partly
like the difference between showing children how to use light-switches and teaching
them about opening and closing electrical circuits. The latter type can explain why
various generalisations hold: a deeper change.

Another common view is that the fact that humans sometimes use diagrams for
mathematical reasoning is just a fact about human psychology, ignoring the fact that
it depends on facts about space and spatial transformations that have nothing
specific to do with human psychology and might be usable by other animals and future
intelligent robots. The facts that some humans can and others cannot use diagrams are
psychological facts, and there are also sociological or anthropological facts about
which sorts of diagrams are used in which cultures. But the study of mathematics is
not the study of such psychological or sociological facts, but the investigation of
deeper truths that constrain what kinds of valid reasoning are possible for any
species, culture, or machine.

Mary Pardoe's proof, above, and the demonstrations in Nelsen's book (see below) are
relevant to investigations of the deeper truths, as I hope will become clearer below.
Perhaps more people building mathematical web demos need to understand the
difference, so that they can help young learners understand the differences between
illustrative examples and proofs/demonstrations.

One of the important facts about such diagrammatic proofs is that their effectiveness
does not depend on the diagrams being drawn with great precision. For examples, the
lines drawn and transformed in the proof, need not be perfectly straight, infinitely
thin or perfectly circular, in order to represent properties of configurations of
perfectly straight, infinitely thin, or perfectly circular lines. That's because the
diagrams, whether drawn in sand, or paper, or merely imagined are not what the proofs
are about: they proofs are about what the diagrams represent (as I argued in 1971).
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Making assumptions of the proof explicit

Attempting to make assumptions explicit invites discussion of an infinite regress of
assumptions, like Lewis Carroll's What the Tortoise Said to Achilles (in Mind 1895).

I believe it is possible to show how to terminate such discussions by showing how to
build a reasoner in which instead of ever more factual assumptions we end with a
demonstration of how something can actually work, and why it works, where the actual
reasoner built instantiates some generic invariances that are equally applicable to
other instances. (E.g. other reasoning animals or machines, whose details can
differ.) But that's a topic for another occasion. For now all I want to do is show
how to make some of the assumptions more explicit than they are in Pardoe's proof.

Unpacking the assumptions of the proof leads to a description of a notion of geometry
that is very close to Euclidean geometry, sharing many features and theorems with it,
though will be slightly different from Euclidean geometry if the parallel axiom is
not provable within it, or if it does not allow the notion of parallelism to be
defined or illustrated. For now, all I want to do is leave open the possibility that
the two geometries overlap without being equivalent (mutually derivable).

I use the label "P-geometry" (for "Pardoe-geometry" or "Process-geometry") to
distinguish this from standard Euclidean geometry. I think there is very substantial
overlap with standard Euclidean geometry, and it may turn out that the two are
equivalent though that is not obvious to me.
Note added: 15 Dec 2010
After writing the above I discovered the chapter by Poincaré mentioned above.
In the light of that, 'P-geometry' could be 'Poincare-geometry'. The book by Jean
Nicod on The Foundations of Geometry, which I dimly recall reading many years ago, is
also relevant.

No doubt there is lots more relevant work, of which I am ignorant.

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Axiomatisations of Euclidean Geometry

Some time ago, for the first time, looked at and pondered the similarities and
differences between Hilbert's and Tarski's axiomatisations for Euclidean geometry, as
presented here: 
    http://en.wikipedia.org/wiki/Hilbert%27s_axioms
    http://en.wikipedia.org/wiki/Tarski%27s_axioms
There are other axiomatisations, but for my current purposes it is enough to (a) to
point out that the same mathematical field (in this case Euclidean geometry) can have
very different axiomatisations and use different initial ontologies and (b) to draw
attention to a few of the differences to illustrate some of the relationships that
can exist between a mathematical field of enquiry and an axiomatisation of the field.

There are several interesting differences between Hilbert's and Tarski's
axiomatisations, including the following: I don't know how many other formal or informal axiomatisations of geometry there are,
nor what is in any of them, so I cannot make any claims about how they relate to
P-geometry. There may already be an excellent axiomatisation of P-geometry that I
have not found.

NOTE:
Insofar as there is agreement among mathematicians that Hilbert and Tarski (and also
others listed on the web site with Hilbert's axioms) produced distinct
axiomatisations for Euclidean geometry, it follows that there is some shared
understanding of what Euclidean geometry is that is NOT defined by a particular
axiomatisation, though that shared understanding provides a basis for generating and
judging axiomatisations.

That raises the question (which has driven my enquiry all along but may not interest
others) what that pre-axiomatic understanding consists in.
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Towards an axiomatisation of P-Geometry

It is worth trying to produce an axiomatisation of geometry that that does not
include the parallel postulate, but does include notions of rotation of line segment
and appropriate axioms corresponding to our intuition that a line segment can
continue rotating, so that its orientation increases or decreases monotonically and
continuously, even if the point of rotation changes, as happens in the Pardoe proof.

In what follows I shall try to indicate how axioms regarding rotating segments and
the angles they sweep through might be used to support something like Pardoe's proof,
without mentioning or presupposing the existence of parallel lines. I shall take for
granted that starting from one of the existing axiomatisations mentioning lines and
segments of a line it is possible to remove the parallel postulate or any postulate
that implies it, and insert the axioms below, to create something very close to
Euclidean geometry. (Later it should be possible to demonstrate this more formally.)

This can be thought of as generalising the notion of a "locus" of a point defined by
some parametrised set of relations, to the "locus" of a changing line segment.

The intuitive notion of such a locus as presented in Pardoe's proof presupposes
change of location in time (hence the rotation). But it may be possible to get rid of
time and just talk about an ordered, continuous, set of line-segments as the locus.

We'll need to refer to the amount (angle) of rotation for part of the locus, and will
have an axiom for rotations something like this:
If the locus of a moving rotating line segment is divided into two or more
parts, each will have an angle of rotation, and the sum of all the angles of
rotation will be the total rotation for the locus.

(This can be generalised to include positive and negative rotations in the same
locus, but that's not needed for the Pardoe proof, since the rotation is monotonic.)

Additional possible axioms are listed below.
This needs to be generalised later to include line segments that not only rotate but
can also be translated in various directions.

It's an open question (as far as I can tell) whether an axiomatisation based on that
assumption (and the further items below) will turn out necessarily to entail the
parallel postulate, or whether this variant of Euclidean geometry is consistent with,
but does not require the parallel postulate.

If the axioms do not entail the parallel postulate, but are consistent with it, then
we'll have a slightly more general type of geometry than euclidean geometry, with
euclidean geometry as a special case. But the theorem about angles of a triangle
summing to a straight line (or half rotation) will still hold.

(Perhaps a meta-theorem is provable that shows that the parallel postulate must
follow from any such axiom set.)

A first draft set of assumptions (axioms) needed for P-geometry, to be combined with
additional standard assumptions for Euclidean geometry, excluding the parallel axiom.
(WARNING: these axioms are likely to be updated. Suggestions for improvement welcome.) Axioms specific to P-geometry Perhaps it will prove possible to transform this to something more elegant and then
show formally whether the intuitive ideas about rotations and translations of line
segments entail the parallel postulate (which is not the same as presupposing it).

I don't know how easy it would be to incorporate these in either Hilbert's, or
Tarski's axioms (after suitable changes to extend the ontology, and removing whatever
is equivalent to the parallel postulate.)

Additional possible axioms -- added 15 Dec 2010

An interesting point is that intuitive notions of space inherently involve time,
since motions are possible in space and motions involve time. Most of the
applications of euclidean geometry in real life presuppose this, and many of the
constructions used in proofs in euclidean geometry are normally thought of as things
that can happen in time. Notions of translation and rotation of rigid shapes and also
some non-rigid shapes with fixed sizes, e.g. inelastic strings, are essential for
engineering.

I don't know if an explicit axiomatisation of Euclidean space with (non-relativistic)
time already exists. But it seems obvious that such a thing is required for Newtonian
physics, and a modified version for relativistic mechanics.

I would not be surprised to learn that some mathematician said all this
long ago.

I think it is close to Kant's philosophy of mathematics, but that's another story.

Added 21 Aug 2013: Origami Geometry

I have just discovered the existence of Origami Geometry which starts from a subset
of Euclidean geometry and adds folding operations. In this geometry it is possible to
trisect an angle:
http://plus.maths.org/content/power-origami#angle
http://plus.maths.org/content/power-origami-0

It is also possible to replace this geometry with a modified version of Euclidean
geometry without folds. 
    Euclidean Constructions and the Geometry of Origami
    Robert Geretschlager
    Mathematics Magazine
    Vol. 68, No. 5 (Dec., 1995), pp. 357-371
    Published by: Mathematical Association of America
    http://www.jstor.org/stable/2690924

Extras
(Needs a section on mathematical education, and the roles of the different proofs.)

Peter Michor made some important-sounding comments on Riemannian geometry and
conformal geometry. I am unfamiliar with both and need to learn about them in order
to understand how they can help.
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Toward robot mathematicians discovering geometry

It will be some time before we have robot mathematicians that understand Pardoe's
proof, or can think about the axioms.

Even longer before a robot mathematician spontaneously re-invents it? (Or the proofs
in Nelsen's book.)

For some speculations about evolution of mathematical competences see

TO BE CONTINUED
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Nelsen's examples of wordless geometric proofs

There is a wonderful book full of wordless proofs that use only diagrams
Roger B. Nelsen,
Proofs without words: Exercises in Visual Thinking,
Mathematical Association of America, Washington DC, 1993,

Viewable on Google Books
Or online as PDF here

It does not include a proof of the triangle sum theorem, though it does include a
proof close to it (by Fouad Nakhli):
14: "The vertex angles of a star sum to 180 degrees"
From the presentation in Nelsen's book, which presents only a star with five vertices
it is not clear how general the definition of "star" is for the purposes of the proof.
The proof given uses parallel lines, and there is nothing to indicate how it could
generalise to a larger number of vertices.

Conjecture:
Assuming that a "star" is defined as a closed, possibly complex polygon formed by
repeatedly drawing lines then turning through an acute angle, always turning in the
same direction, then it seems that the method suggested by Pardoe, using monotonic
rotation of a single line segment about several points in the plane will prove a
general form of the star theorem, provided that the number of vertices is odd.
__________________________________________________________________________________________

Acknowledgements

In addition to Mary Pardoe, I owe a debt of gratitude to the various people who
commented at various stages of the discussion, helping me to clarify what I wanted to
say by pointing out flaws in what I actually said (at first).

None of the following should be taken as agreeing with anything written above.
However their comments, both critical and friendly helped to produce whatever results
there may turn out to be in the rest of the document.

Thanks to: Arnon Avron, James Davenport, Manfred Kerber, Tom Leathrum, Paul Libbrecht,
Peter Michor, Dana Scott, and of course Mary Pardoe who started it all.

Tom Leathrum has a collection of useful Java applets for mathematical exploration by
students: http://cs.jsu.edu/~leathrum/Mathlets/
I have only sampled a small subset.

NOTE:
I am aware that there has been a great deal of work on geometry about which I know
nothing and that it is very likely (a) that there is nothing really new in this
paper, and (b) if there is anything new it is likely to turn out to be either
mistaken or not as well formulated as it should be.

Offers of help in making progress will be accepted gratefully, especially suggestions
regarding mechanisms that could enable robots to have an intuitive understanding of
space and time that would enable some of them to rediscover Euclidean geometry,
including Mary Pardoe's proof. I believe that could turn out to be a deep vindication
of Immanuel Kant's philosophy of mathematics. Some initial thoughts are in my online
talks, including 

http://www.cs.bham.ac.uk/research/projects/cogaff/talks/#toddler
Why (and how) did biological evolution produce mathematicians?
__________________________________________________________________________________________

Maintained by Aaron Sloman
School of Computer Science
The University of Birmingham