It is not clear to me whether the result turns out to be a superset of Euclidean geometry. It seems not to involve the parallel postulate, and therefore to be a different geometry, unless the parallel and the new postulate can be derived from each other.
NB: This is still work in progress, and likely to be changed/extended.
Installed: 9 Dec 2010
Installed and maintained by Aaron Sloman
Last updated: 25 Aug 2016; 8 Sep 2017
10 Feb 2016: Added inverse rotations and an axiom for "reverse rotations" that adds the Euclidean property [below].
29 Mar 2015: Added link to discussion of angle trisection and Poincaraé's philosophy of mathematics. Some reorganisation and re-formatting.
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
A partial index of discussion notes in this directory is in
See also this discussion of "Toddler Theorems":
Mathematical phenomena, their evolution and development
(Examples and discussions on this web site.)
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
(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:
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.
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.
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:Two "standard" proofs of the triangle sum theorem using parallel lines are
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
The page teaches concepts with some interactive illustrations, but presents no proofs.
Warning: several online proofs seem to have bugs due to carelessness.
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.
An alternative presentation as a process
Available as printable PDF here.
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.
From the ensuing discussion on the email list I was able to distil the following alternative views. (Participants and others are invited to comment on this summary.)
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.)
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.)
I append the work in progress below, in the form of a partial axiomatisation.
In issue 73 (2010) 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
NOTE: I discussed both of these points in my DPhil thesis and in various
papers and presentations since then, e.g.
In Principles of Knowledge Representation and Reasoning:
Proc. 5th Int. Conf. (KR `96),
Eds. L.C. Aiello and S.C. Shapiro,
Morgan Kaufmann, Boston, MA, 1996, pp. 627--638,
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)).
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).
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.
There are several interesting differences between Hilbert's and Tarski's axiomatisations, including the following:
Rotations, but not translations, are used in the Pardoe proof displayed above. There is another version of her proof, which alternates between rotations and translations, to ensure that the rotating segment rotates only about one of its ends available here.
I suspect there are more variants, or generalisations, of this proof.
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 a question, which has driven my enquiry all along but may not
interest others, namely what that pre-axiomatic understanding consists in.
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 needs to be generalised later to include line segments that not only rotate but can also be translated in various directions.
(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.
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.)
Lines have positions and orientations in the plane. line segments have positions and orientations on the plane.
Operations produce changes of positions and orientations.
A path for an entity is an ordered, continuous, complete, set of positions and orientations for the entity. This needs to be unpacked, to define the notions of continuity, ordering, etc.
Some first draft suggestions
A set of entities with an ordering is "continuous" (weakly continuous?) if given any two different items in the set there is at least one more that is between them.
The set is "inclusive" (is there a better term?) if it contains all the items that are between any two that it contains.
(For present purposes we do not require full mathematical continuity.)
[... other general axioms are needed, e.g. about orderings of positions and orientations ...]
The position of the line segment in the plane is also the position of the segment on the line containing it.
In a cyclic translation the initial and final positions are identical.
If no two positions in the set are identical the path is monotonic.
A simple rotation defines a rotation path in which all the lines containing the positions of the segment have a unique point in common.
That common point is the "centre of rotation" of that rotation.
If no two positions in a simple rotation are identical the rotation is monotonic.
If the start and end positions in a simple rotation are identical, and only those two, the rotation is a cycle.
If a simple rotation of a segment is monotonic, but the start and end positions of the segment lie on the same line, but with the order of the ends on the line reversed, then the rotation is a half-cycle.
A compound rotation defines a rotation path in which there is no point common to all the lines containing the positions of the segment.
A stepped rotation of a segment is a compound rotation whose path is composed of an ordered set of sub-paths each of which is simple.
Each subpath of a stepped rotation, except the last, ends with a position that is the start position of the next sub-path.
(Mary Pardoe's proof above, uses a stepped rotation, using the vertices of the triangle in turn as centres of rotation.)
(To be continued: Need to define cycle and half-cycle for compound rotations.)
This is also the difference between the orientations of the lines containing the initial and final positions of the segment.
(What properties do we need to assume for angles. Do they have to be scalar (metrical) values or could they form a partial ordering?)
NOTE: This assumes that the segment is rotated about a point on the line containing the segment. Allowing rotations about a point not on the line L adds extra complications, since the segment will then be constantly translated as well as being rotated, and the points common to L and the intermediate lines containing S during the rotation will all be different.
The parallel postulate could be brought in by adding an axiom to allow rotation about a point not on L and then postulating a unique position during the rotation when the line through S no longer intersects L.
I don't see that we have to include this possibility.
Such an axiom could define a special case of P-geometry. But it may be required all angles anywhere in the plane are to be comparable -- i.e. forming a total ordering (conformal geometry?)
P-05-dud: The original formulation of this axiom did not allow for a segment to be rotated until it lies on the same line as before. The axiom originally stated: when a line segment is rotated there is always exactly one point common to the original and final positions: the point of rotation.
(We can define acute, obtuse and reflex angles).
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
The next time S returns to L, the sides will be where they were originally. (A full-rotation.)
After a full rotation the situation is indistinguishable from the initial situation.
In the latter case there will be a discontinuity between the initial state, when every point of S is on L, and the set of subsequent states, when no point of S is on L.
Do we need an axiom stating that if the lines containing two segments have no point in common then there is an infinite set of additional line segments between them whose lines have no point in common. (I.e. an infinite plane between two non-intersecting lines can be swept out by a moving line.
Need to define perpendicularity (half rotation) and the sense of a rotation. If two rotations of opposite sense with the same initial segment and the same point of rotation are equal then if the final states form a straight line the "shared" edge is perpendicular to the line. S[perp]L,
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.
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 discuss below a version of one of these articles, namely the one on Non-Euclidean geometries.
The following extract is directly relevant to our discussion.
This seems to be related to the method used to trisect an angle referenced below.
Foundations of Geometry and Induction
By Jean Nicod
With an introduction by Bertrand Russell
Available on google books
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 See the discussion here: http://en.wikipedia.org/wiki/Angle_trisection
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
Graham Nerlich, 1991, How Euclidean Geometry Has Misled Metaphysics, The Journal of Philosophy,It points out that a common philosophical argument, namely that if everything were to gradually double in linear dimensions during a period of time that would be undetectable, and therefore there is no such thing as absolute size, breaks down if space is non-Euclidean.
88, 4, Apr, 1991 pp. 169--189,
This has implications for the status of the Pardoe proof and also the status of P-geometry. I may later add some comments about that here.
Roger B. Nelsen,It does not include a proof of the triangle sum theorem, though it does include a
Proofs without words: Exercises in Visual Thinking,
Mathematical Association of America, Washington DC, 1993,
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.
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.
In an email discussion 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.
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.
Andrea Asperti, Arnon Avron, James Davenport, Michael Fourman, 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
I have sampled only a small subset.
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.
Since September 2020 I have been thinking about hatching processes in the eggs
of vertebrate species that emerge from their eggs not only with complete,
functional, highly complex, physiological structures, but also spatial cognitive
competences that they use too soon after hatching to have had time to acquire
them by learning (e.g. training neural networks). The analysis of requirements
continued during 2021 and 2022 and eventually produced a complex theory about
interactions between evolution and developmental processes in eggs. A summary of
some of the issues and my still evolving ideas can be found in this document,
and others it refers to:
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 related thoughts are in my online talks, including this old talk:
Why (and how) did biological evolution produce mathematicians?