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It only takes a minute to sign up. I am teaching a course in Euclidean geometry at the University of South Carolina, and it seemed highly appropriate and interesting to read Euclid himself. See here for a wonderful, and completely free, translation and guide. We are up through Proposition 17 now, and although this is very instructional for both the students and myself, I sense that before too long the novelty will wear off and it will be good to return to a modern treatment. I will, though, want to say a little bit about his treatment of parallel lines and elliptic and hyperbolic geometry.

That said, there are some gems. In Book 2, Euclid constructs square roots, and in Book 4 he describes how to draw a regular pentagon which is surely not obvious. And, the constructions in the first few books are all very interesting.

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Of course there is lots of fascinating number theory too, but my course is on geometry. What other particularly fascinating tidbits do the Elements contain, which it may be easy to overlook? If I were to ever incorporate Euclid's elements in a class, it would be through a critical exposition of Euclid's work, not necessarily directly with his books. The best source I've gotten into for that is Hartshorne's Euclid and beyond.

It really helps you see the importance and impact of Euclid's work, without really having to trudge through it. As I recall, the gems are strewn along the path of good exposition. The book also introduces Hilbert's axioms, which was the next major revolution of the foundations of geometry that students should really hear about.

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It never lingers on Euclid's presentation long enough to become boring. It revisits it throughout the book, of course. Really, there is no reason to linger on Euclid after students have gotten the general feel of the material. For me, I like Book 1 Prop 35, whichis the first use of "equal" to mean equi-areal rather than congruent. Book 2 Prop 11, where Pythagoas is used geometrically to prove a construction for the golden ratio later used for the regular pentagon construction again a geometrical use of Pythagoras to prove the "power of a point" showing that the angle bisector cuts the base of a triangle in the same ratio as the other sides.

Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Hidden geometrical gems in Euclid's Elements? Ask Question.

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Asked 6 years, 11 months ago. Active 5 years ago. Great question! We just spent a school year with a geographic atlas that had been meticulously represented in raised-line graphics but was for the most part extremely difficult for my daughter to use. She needed extensive one-on-one instruction with a sighted person who had the print atlas in hand in order to do her assignments, and it took many many hours.

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Of course, the school didn't provide that person--it was dad. Thanks for your good work,. This is not a graph or a pie chart or some other 2-d visual illustration. This is effectively creating an optical illusion through a tactile medium.

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How would you propose creating an optical illusion for someone who has no concept of the subtle effects of light on the world around us? It may not be an identical exchange; like you say, visual processing is suggestible in ways that touch, perhaps, is not. Unfortunately, some educators believe that, since those senses are not equivalent, there is a commensurate lack of ability to understand certain concepts, and thus blind students do not have to be taught those concepts. The only time my child is provided objects rather than drawings often poorly-rendered and instruction is when her dad or I provide them.

I have seen on Math Olympiad problem sets the type of test question being referred to.

ncof.co.uk/despus-de-la-niebla.php The problem does not concern visual illusions per se, but rather spatial conceptualization. Vision is not the only way to understand spatial concepts. In the US, what is taught is often driven by what is on the standardized or criterion-referenced test at the end of the year.

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If the testing agency simply deletes any question that concerns spatial concepts, or provides poor representations of it, then blind students will not be taught the content represented by that question. Ideally, the blind student should be given the object to study and to use on the test--why is the drawing better than the real object anyways? And yes, having written my doctoral thesis on eye development, I do understand something about the evolution of visual processing.

Sorry to everyone else on the list; I am feeling provoked. Besides, picking on a part of what I had to say is a bit disingenuous.

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I ask you again, how can you create optical illusions through a tactile medium? Let me help you, you can't! As I've already pointed out, teaching the underlying principles is far more important than wasting time on wretched diagrams. A firm grasp of the abstract concepts will actually help the child develop his spacial awareness.

What about those kids whose parents aren't aware of the shortcomings of math education or who themselves think their kid can't do geometry? Somehow I doubt you have the courage or the decency.


Save me from hypocrites! But how many kids actually get that kind of experience? Some kids get the manipulatives but a lot don't. Even sighted kids would benefit from the manipulative because everyone learns differently. The thing with testing is that most developers don't get it that if they want to test fairly, then they need to find out what is fair.

Some of the tactile graphics I've seen are horrendous and they "pass" because they look pretty to a sighted person. Nelson Blachman nelson. Perhaps the blocks representing the steps of a stairway should grow smaller and smaller from the bottom where the viewer may be standing to the top, and perhaps the blocks of equal size we've been discussing represent a false, unhelpful idea of perspective. Maybe it would be better to learn than to pass the test.

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Sharon Clark sharonjackson03 comcast. I have also seen them represented using various line textures. It would be nice that the students have similar drawings in their math textbooks, but this is usually not the norm. I, as a teacher of the visually impaired, who is also blind myself, try and find various representations of the same drawing concepts so that the student may be familiar with whatever statewide test drawings are given.

It is nice when the developers take the time to accurately represent a drawing in the tactile form, but it also depends on the experience of the student. I, myself, understand both types of 3d drawings using the two methods mentioned above, but I was exposed to both. It is extremely frustrating to see poorly designed tactile drawings in mathematic textbooks due to inexperience of the developers. What little vision I do have is only in 1 eye or the other, not both.

I only use 1 eye at a time so I have no depth perception. Susan Jolly easjolly ix.