Does this forum include euclidean geometry?

[mathwonk]

I am teaching euclidean geometry this fall and realized i don't know it that well. there are some famous modern versions of the axioms which do not completely satisfy me, such as hilberts, gasp. i said it.

i especially like the new book by hartshorne, geometry euclid and beyond, because he makes it so clear that i can choose to disagree with him and still he covers my point of view too.

e.g., i think it less natural to choose pasch's theorem as a postulate (a line meeting a triangle away from a vertex meets it again.) than simply to say a line separates the plane into two disjoint sides.

i also think it unintuitive to postulate that SAS implies congruence as so many books do now in high school, such as the previously wonderful book of jacobs.

it seems more in keeping with euclid to postulate instead a set of rigid motions, and use them as he did to prove SAS as a theorem.

I mean I think the most natural axioms are euclids stated ones plus his unstated ones that he used in his proofs. this to me is genuinely euclidean geometry

fortunately hartshorne proves that one can substitute rigid motions for the SAS postulate he himself uses, and it is easy to show that Pasch can be replaced by plane separation.

but i suggest high school books could benefit from such changes too.

on the other hand high school books actually do not always even mention the fact that a line separates the plane into two sides, as if it is better left unsaid.

without it however, one might be working in three space! i am taking an intuitive approach to geometry at first and hoping to gradually introduce more precise language and proof at the middle or end.

do you recall such theorems as the incidence results for medians, altitudes, perpendicular bisectors? angle bisectors? and their link with inscribing or circumscribing circles about triangles? does everyone (except me) know that the angle cut by a pair of rays with vertex on a circle, does not depend on the location of the vertex? just of the two other intersection points with the circle?

i.e. any pair of rays with vertex on a circle subtends an angle of exactly half the arc cut from the circle by the two rays. more people probably know that any angle with vertex on a circle, and with rays cutting the circle at the ends of a diameter is a right angle.

it is also easy without calculus to compute both surface area and volume of a sphere, (finessing limits), using cavalieris principle. it seems actually easier and clearer to explain them this way than to use calculus.

im beginning to find this stuff interesting. then too the contrast betwen euclidean and spherical and hyperbolic geometry is interesting.

what do you guys think?

 

[Nancarrow]

Wow. Do American high schools teach Euclidean geometry with a postulational approach?

That's so cool. I'm from the UK and I'm pretty sure that back at school, geometry was mostly just a set of rules to learn and apply. My first contact with Euclid was when I bought the books a couple of years ago (I'm 28).

 

[symbolipoint]

Wow. Do American high schools teach Euclidean geometry with a postulational approach?

Yes, they definitely do. That kind of Geometry was going out of style a few years ago but is returing fully in several places.

That's so cool. I'm from the UK and I'm pretty sure that back at school, geometry was mostly just a set of rules to learn and apply. My first contact with Euclid was when I bought the books a couple of years ago (I'm 28).

The well-designed courses of college preparatory Geometry are concerned with basic non-definable notions, postulates, theorems, constructions, problem solving, proofs, and creating written explanations.

 

[Gib Z]

mathwonk: "does everyone (except me) know that the angle cut by a pair of rays with vertex on a circle, does not depend on the location of the vertex? just of the two other intersection points with the circle?"

A more well known way of stating the theorem is: Angles standing on the same arc of a circle are equal. At least, that's if I understood your quote :(

And I'm in high school..and I did learn SAS implies congruency :( Why doesn't it >.<

 

[AlephZero]

Do you recall such theorems as the incidence results for medians, altitudes, perpendicular bisectors? angle bisectors? and their link with inscribing or circumscribing circles about triangles? does everyone (except me) know that the angle cut by a pair of rays with vertex on a circle, does not depend on the location of the vertex? just of the two other intersection points with the circle?

i.e. any pair of rays with vertex on a circle subtends an angle of exactly half the arc cut from the circle by the two rays. more people probably know that any angle with vertex on a circle, and with rays cutting the circle at the ends of a diameter is a right angle.

Yes - and they practically quite useful results, at least for mechanical engineers.

As a side comment, it's interesting (to me anyway) that there were some "elementary" incidence theorems that were missed by the Greeks, and not proved till the 19th century - though there's no obvious reason why the Greeks couldn't have proved them. Desargues theorem, for example. Though paradoxically it's easier to prove it in 3-D than in 2-D.

AFAIK the Greeks never discovered the 9-point circle of a triangle either - see http://www.cut-the-knot.org/Curriculum/Geometry/SixPointCircle.shtml.

When I was a lad, we did stuff like the 9-point circle in school geometry at age about 15. We didn't get hung up about the philosophical implications of alternative axiom systems, though!

 

[symbolipoint]

The topic's question may have the answer: Geometry can be discussed in the Mathematics General board. It could as well be part of "PreCalculus".

 

[mathwonk]

the proof of SAS by euclid uses the orinciple of "superposition", i.e. the fact that one can move triangles around in the plane by translations rotations and refklections without changing distances. this fact was niot assumed by him as a postulate, so either you have to assume this fact or you have to asume the SAS theiorem as a postulate.

different books begin from different perspectives, andsome introduce real numbers and some do not, as eucklid did not have them.

for instance, to say that two segments are congruent you can either say that means there is a translation and rotation taking one onto the other, or youm can say there is a ruler available to emasure their lengths and they have the same length, or you can say congruence of segments is an undefined term, with certain postulated properties.

i must admit though that often in the US, high school geometry is merely a list of formulas like A = pi R^2, and they just plug various values of R into this. no theorems, no postulates, nothing of value at all.

 

[tronter]

not in my school. We studied euclidean geometry rigorously (2 column proofs). We learned many proof techniques, and theorems.

 

[Doc Al]

i must admit though that often in the US, high school geometry is merely a list of formulas like A = pi R^2, and they just plug various values of R into this. no theorems, no postulates, nothing of value at all.

Things must have changed since I was in HS. (Quite a while ago, I will admit.) Our geometry was all postulates, theorems, and proofs--I loved it.

 

[mathwonk]

education in US is very diverse, i live in GA

 

[jim mcnamara]

Mathwonk - My condolences on your local schools...

In grade seven, almost sixty years ago, we learned the first four postulates. Spent most of the year on them. When we got to number 5, it really went sideways... We started with the Playfair posulate (axiom). Then we heard about Bolyai and the fact that you can make different geometries by diddling that one postulate - turning things into elliptic and hyperbolic geometries.

Sorry to say, we learned Hilbert's 21 (now I think it is considered to be 20) axioms, too... My teacher thought his take on any topic was the bee's knees. She got her degrees in the twenties. So she literally used that phrase.

She also said he was the last mathematician to have fully understood all of extant Mathematics -- as it existed during his time. I do not know if that is correct or not, but it sounded great back then.

 

[MathematicalPhysicist]

not in my school. We studied euclidean geometry rigorously (2 column proofs). We learned many proof techniques, and theorems.

me too, and i live in israel, 2 columns proofs i.e an argument and justification of it.
but I must admit that i don't rememeber a lot from it, 5-6 years has passed since, I rarely used it in my classes in univ, but i think that in 2008, i think i could take the course non euclidean geometry, which is more modern than the approach we had in high school.

 

[Chris Hillman]

She also said [Hilbert] was the last mathematician to have fully understood all of extant Mathematics -- as it existed during his time. I do not know if that is correct or not, but it sounded great back then.

Sure she didn't say that about Poincare? (See the book by E.T. Bell, Men of Mathematics.)

Mathwonk, are you by any chance teaching geometry for future high school teachers? I think these courses should include a lot of transformation geometry. I presume that Martin, Transformation Geometry: an introduction to symmetry, Springer, 1982 is an example, but right now I can't seem to recall having examined that book--- I was looking for another book with a similar title. Coxeter, Introduction to Geometry, Wiley, 1969 is a really good textbook for a good "geometry for teachers" course.

 

[MathematicalPhysicist]

Sure she didn't say that about Poincare? (See the book by E.T. Bell, Men of Mathematics.)

Mathwonk, are you by any chance teaching geometry for future high school teachers? I think these courses should include a lot of transformation geometry. I presume that Martin, Transformation Geometry: an introduction to symmetry, Springer, 1982 is an example, but right now I can't seem to recall having examined that book--- I was looking for another book with a similar title. Coxeter, Introduction to Geometry, Wiley, 1969 is a really good textbook for a good "geometry for teachers" course.

why can't he teach simple minded undergrad, or ones that are heading to graduate school?

 

[mathwonk]

iam trying to teach simple minded geometry, and then some insight into less simple minded approaches to it.

 

[robphy]

"Experiencing Geometry"
https://www.amazon.com/dp/0131437488/?tag=pfamazon01-20
sounds like an interesting book [supplement]... but I haven't seen it in detail.

 

[Gib Z]

the proof of SAS by euclid uses the orinciple of "superposition", i.e. the fact that one can move triangles around in the plane by translations rotations and refklections without changing distances. this fact was niot assumed by him as a postulate, so either you have to assume this fact or you have to asume the SAS theiorem as a postulate.

Im confused..is it just the Euclid's proof incomplete? Or is there some counter example we have where SAS isn't proving congruency..perhaps this is true for the Euclidean plane only?

 

[mathwonk]

review of harold jacobs 3rd edition

This is indeed a wonderful geometry book for high schoolers, but not in my opinion quite as wonderful as the first edition. Visually it is more colorful and more appealing, but I think the content is in some ways more solid, and in some ways less so.

As a book for any high school course, this would make a wonderful choice, but to me its usefulness could be enhanced by some discussion of why things are done as they are, how they could be done otherwise, so the teacher can make some independent choices, and the student can be less puzzled in some cases.

I.e. there are basically 4 ways to present geometry,
1)as a catalog of true facts to be discovered about the usual real number plane R^2 of calculus and precalculus courses. This is the default mind set of a an average student in my guess, i.e. the Euclidean plane does exist, and we are trying to learn things about it.

2) A modification of the previous approach does not assume the Euclidean plane is covered by a grid of coordinate lines, i.e. does not assume it is actually equal to R^2, but does assume each line in the plane has a ruler, i.e. that there are linear coordinates on each line, and angular coordinates on each semi circle. These ruler and protractor postulates allow one to transfer facts about real number into facts about lines and angles. This is the approach (due to G.D. Birkhoff) taken by Jacobs in this book. Congruence of segments and angles is taken to mean equal measure as real numbers.

Now certain facts about the plane have to be taken on faith, i.e. as postulates, such as SAS, a theorem in Euclid which Jacobs assumes as a postulate, with no discussion of why it is no longer a theorem. In fact he also assumes ASA, and in the first edition he assumes also SSS, but in the third edition he proves SSS as a theorem. The natural question of what is going on here? is never addressed. Moreover, since the students do not have a firm grasp of real numbers, and Jacobs does not discuss them. Merely referring to distances as "positive numbers", their properties cannot truly be transferred over.

3) A third approach due to Hilbert, is to ignore real numbers and present a purely geometric development of geometry from which one can then deduce the existence of real numbers, and also of other more subtle fields of numbers. He postulates everything needed, separation properties, SAS, ... Here even congruence is not defined, but merely understood as an undefined term whose properties are given in the axioms.

4) Another approach would be to postulate the existence of rigid motions, something Euclid took for granted in his own proofs, i.e. the method of super position, or moving figures around by translations, rotations, and reflections, without changing lengths or angles. I like this approach, as it recovers Euclid's own intuitive approach to the theorem SAS, and restores it as a theorem, not an axiom. This approach is taken in the classic French text by Hadamard.


To see the difference in these approaches, consider the proof of SSS, in Jacobs 3rd edition, page 164. This is a very nice proof, adapted from Euclid, merely by making Euclid's proof positive instead of negative in approach. I.e. this is the contrapositive of Euclid's proof, by the same method, using isosceles triangles.

But note Jacobs himself apologizes for the length of the proof. If you consult Euclid you see it was not originally this long, so what gives? It is the trivial first step of the proof that has been drawn out by Jacobs insistence on ignoring the principle of superposition, i.e. of admitting rigid motions exist, as every child would do.

Thus Jacobs has to resort to unnatural and tedious applications of various ruler and protractor postulates just to move one triangle over onto the other one, so that their congruent bases coincide.

If one accepts this one fact as an axiom, instead of the less obvious SAS (which of course says that caliper measure coincides with tape measure), it is easy then to prove SAS, SSS, ASA, and AAS, all by the same picture.

So by not clarifying the properties of numbers that will be used, Jacobs gives an illusions of rigor that is not justified. He also apparently passes over all of Hilbert's and Birkhoff's separation postulates without mention, which is probably well advised for high school. But notice that without the separation postulates, the claim of Jacobs on page 226, that geometries where the parallel postulate fails were only found in the example of exotic non Euclidean geometries, is not true at all, since all of Jacobs previous postulates are true in R^3! I.e. he has no postulate before postulate 7 on p. 742 that does not hold in R^3.

Now here is another little remark where Jacobs confuses two different ideas, that of rigidity and that of congruence, on page 163, where he claims that the rigidity of a triangle implies that SSS implies congruence. This is simply false. Rigidity of a figure is a local property, and congruence is a global one. I.e. a given property of a figure is rigid, if any nearby deformation of the figure changes those properties. But there is no guarantee a rigid property may not be shared by another figure which is obtained by a large, but not a small, variation of the given one.

E.g. the property of ASS, i.e. fixing two sides and an angle not contained between them is also a rigid property, since any nearby deformation of a triangle will have one of those three measures different. But those three measures, although they rigidify the triangle, do not determine it, since there usually is another non congruent triangle with very two different angles, but still having the same ASS of the given one.

Finally, when Jacobs does introduce rigid motions, he states that congruence by motions is clearly equivalent to congruence by numerical measure, pages 99, 140, 312-3, which is not obvious at all. It is not even obvious that a reflection preserves collinearity without some proof.

In the first edition of Jacobs, these same errors and assertions existed, but there was a more thorough discussion of logic and proof, and the book seemed to me aimed at a more perceptive reader than the third edition.

I conclude that the book is truly excellent but has flaws, some of which I have trued to make visible for eraders who may wish to choose alternative treatments, such as that in the book from the University of Chicago project, by Coxford, Hirshorn, and Uziskin, or the little book by Clemens and Clemens, Geometry for the classroom, or even the original book of Euclid.

It was a shock to me to realize today that Euclid (in the beautiful Green Lion press edition) is much more readable than many more recent versions.

 

[mathwonk]

a "proof" is a logical deduction of certain statements from others. it is not possible to deduce SAS just from the axioms euclid gave.

i.e. there exist model geometries where all his postulates hold but SAS does not. Of course these models are not euclidean planes, which again says that euclids axioms do not suffice to determine the euclidean plane as we know it, namely R^2.

i.e. SAS is true in R^2, and it is true in any geometry satisfying certain larger collections of axioms than euclid gave. e.g. if we assume you can move triangles around without changing the size of their sides and angles, then SAS is true, and Euclid proves it.

Actually he needs a few more unstated things too. like every line in the plane has just two sides, etc... but we skip over these.

Euclids proof shows that if two triangles share a base, i.e. have physically the same base, and also the two angles adjacent to the base are pairwise equal in the two triangles, then the triangles are congruent.

But he does not prove, and cannot prove with only his postulates, that if a triangle has a base congruent to the base of another triangle, then we can move the first triangle over so that its base actually coincides with the base of the other one.

he took this for granted. so if you want to enhance his postulates, you either have to assume rigid motions exist, or have to assume SAS as an axiom, or something else equivalent.

so one way to prove SAS in R^2, is to discuss the orthogonal matrix group O(2), of length preserving matrices whose columns are orthonormal bases of R^2, plus translations, and then by combining them you have enough rigid motions.

 

[mathwonk]

imagine a bumpy rug, where two points in one place are in a flat part of the rug, and two points in anoither place have a bump in between. Now take a compass and set it so it measures the first two points exactly. Then assume it also measures the other two points exactly.

But if we try to measure the distances uing a tape measure, the bum makes the second pair of points look further apart. I.e. SAS says that tape measure distance agree with caliper, or compass, measured distances, and this says your space looks the same in all parts of space, something euclid never assumed.

e.g. it appears to be the same crow fly distance from seattle to tacoma as from seattle to bremerton, i.e. from a satellite they look the same, but you can't tell that with a car odometer because puget sound lies between seattle and bremerton.

 

[mathwonk]

it sems that it is usual to assume SAS as a postulate in a geometry with rulers and protractors, and deduce SSS, and other congruence results.

But it sems to be unknown, as of 15 years ago, if one can assume instead SSS, and deduce SAS. Is there news on this front?

 

[symbolipoint]

A great, modern Geometry textbook presents SSS, SAS, and ASA facts for triangle congruence as postulates; and then presents AAS as a theorem and provides the proof.

 

[mathwonk]

well harold jacobs does that in his first edition. but these postulates are logically excessive. i presume he assumes the three postulates for pedagogical reasons, so as not to burden the reader with too many proofs. but it is known that one can logically get by with only the one assumption of SAS, proving all the others.

Jacobs himself changes his postulates in the third edition of his book, assuming only SAS and ASA, and then proving SSS and AAS as theorems.

What I am asking if whether logically one can get by with just assuming SSS, and can then prove the others?

 

[mathwonk]

it seems the best book for euclidean geometry is euclid. to see this, just read it.

 

[mathwonk]

I will expand on that last remark:

Euclid has 5 postulates, and a few unstated ones, like existence of rigid motions and the fact that lines separate the plane, and that lines have the archimedean property.

but he tried to proceed as far as possible using only the first 4 postulates. hence he proves essentially correctly, (provided you actually state his unstated assumptions), all the usual triangle congruence theorems, SAS, SSS, ASA, AAS, without using the 5th postulate. (OOps, his first propositiion needs the fact that two circles do intersect if their centers are closer than the sum of their radii).

But this means these theorems are all true also in hyperbolic geometry. I.e. his first 28 or so propositions are theorems in what is now called neutral geometry.

Not having read Euclid, I gave proofs in clas that used purely Euclidean proofs, e.g. using the angle sum of a triangle to be 180 degrees, or using the existence of rectangles, or the pythagorean theorem.

then i read in millman and parker the proof of AAS without using the fifth postulate, and they bragged that the reader had probably never seen this proof before, which was true.

But THEN I opened Euclid, and found that they were just copying his original proof! I.e. I learned that the carefully neutral beginning of geometry study was due to Euclid himself, and was the original way to present it.

So I went back and represented yesterday all the previous triangle results as Euclid himself had done, so the clas could see clearly which thigns could be done without the 5th postulate.

next we will do what saccheri did, i.e. go a bit further than euclid. Saccheri noticed that a few more facts could be proved, still without the 5th parallel postulate, in particular that the exterior angle of a triangle is greater than either remote interior angle, a triangle has at most 180 degree angle sum, and that a quadrilateral with two equal and opposite sides perpendicular to a common base, has a top which is at least as long as the base, and top angles which are equal and at most 90 degrees. one can also prove the triangle inequality that two sides of a triangle together are longer than the third, and in a right triangle any two sides determine the third.Then we will at last introduce the 5th postulate exactly as Euclid did, and begin to show how one can get stronger resuts, such as the angle sum in a triangle is actually equal to 180, the saccheri quadrilateral is a rectangle, pythagoras holds,...probably we won't have time to pursue the consequences of assuming the 5th postulate is false, i.e. learning facts in hyperbolic geometry, e.g. that similar triangles are also congruent!...

But we are having a good time, and I am finding again that having tried to get down on a beginning level to actually explain the material from scratch, I am understanding it myself for the first time!

Because of this experience, I recently purchased the translated works of Saccheri, Euler, and Archimedes, (in addition to Euclid), but have not had a chance to read from them yet.

 

[mathwonk]

wow, just opened archimedes and found precisely the statements i have been making to people lately about circles and spheres:

namely he says exactly that (for the pruposes of computing area) a circle is a triangle whose vertex is at the center and whose height is the radius. and (in regard to volume) a sphere is a cone whose vertex is at the center and height is the radius.

it makes me fel very good to have arrived at exactly the perspective of archimedes in these matters, although 2,000 years later, and with a great deal of instruction.

too bad i did not read him as a child, i would have known this 50 years ago.

oh, and archimedes knew the location of the centroid both of a triangle and of a cone, the latter something i just learned a few weeks ago and taught to my class.

and euler computes the values of zeta at even integers, i.e. the sum of the reciprocal powers 1/n^2k, for k up to 13, in his precalculus book, vol 1, page 139. actually he just states them, implying the computations are straighforward.

i have still not arrived at euler's perspective.

 

[mathwonk]

i have a geometry question.

i am teaching geometry now and trying to prove as much as possible without using the parallel postulate. we have the congruence theorems, SAS, SSS, ASA, AAS, and the weak exterior angle theorem, open mouth theorem, hypotenuse - side congruence for right tringles, and the fact that the angle sum in a triangle is never more than 180degrees, and we are ready for the concurrence theorems for angle bisectors and medians.

(of course one knows the other two basic congruence theorems, for altitudes and perpendicular bisectors, do not hold without the 5th postulate.)

but i cannot find a book with a proof of these things, or at least not for the medians. the books i have, like greenberg, prove first that there are only two geometries satisfying ll these axioms, short of the parallel postulate, and then give two different proofs of the concurrence theorem ffor medians, one for euclidean geometry and one for hyperbolic geometry, using the poincare model.

now clearly that is not the right way to proceed. one should give a single unified proof just using the axioms that hold for both. any suggestions for a reference that does it right? i have a kind of a vague idea myself but do not know how to implement it.

(i wonder if ceva's theorem works in this generality?)

 

[mathwonk]

yes that seems to do it. so greenberg does apparentkly have the proof, but the reference i got from millman parker is to the wrong oart of greenberg.

i.e. the proof of ceva in Greenberg, exercise H6 p. 234, is apparently implied there to work in general using the theorem of menelaus, to prove that the medians are either concurrent or parallel, but it is easy to show they are not parallel.

 

[mathwonk]

oops, the proofs of cevas thm i am finding use unique parallel lines, or similasrv triangles or areas, which are not elementary concepts (or not true) in neutral geometry.

 

[mathwonk]

ah yes, the hyperbolic area function does NOT satisy bh/2 for area of a triangle, so the various area proportionality proofs of ceva do not hold. in fact the numbers bh/2 are not the same for different choices of base and height.

presumably ceva is true however??