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Geometry
Subjects:
- Suggestions for a raised drawing of stairs made of cubes (blocks)
- A Regular 10-faced Solid and The Cube
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1. Subject: Suggestions for a raised drawing of stairs made of cubes (blocks)
gpatterson002@mchsi.com
Fri Jun 6, 2008
I am proofreading a Braille copy of a test given to students here in Iowa. One of the questions asks how many cubes are in the drawing of stairs made up of blocks. So, I believe the generic problem is how to represent a three-dimensional figure in a raised drawing such that a Braille reader can solve this "simple" problem. The testers are willing to drop the questions presenting this type of image. The Braille transcribers have produced the raised drawing exactly as the figure is shown in the print copy. I haven't thought of any way to present this image so that a Braille reader can distinguish the number of blocks composing the stairs.
Any suggestions would be nifty. In advance, any solutions would be appreciated.
Responses:
Matthew2007 matthew2007@charter.net
Fri Jun 6 2008
HI
Since it appears as though the concept the test is trying to assess is one of a depth perception visual spatial type, how about using thickness in the lines of the drawing as a method of conveying the ascendance of the blocks. You would start out with blocks with thick lines at the base and gradually decrease the thickness of the lines used to draw the blocks as the placement of the blocks gets higher and higher as the structure grows until you eventually end up with the top layer of the blocks having very thin lines to denote their upper placement in the structure.
You might also add different textures to the blocks to differentiate the various levels of the blocks. The first bottom layer might have an intense grainy "coloring" while the blocks above that level might have a less intense feel/coloring. The top layer might have only the slightest texture to completely differentiate it in comparison to the bottom layer of blocks. In other words, you are taking the inherent 3D aspect of the 2D picture and finding a manner of creating depth perception for the blind test taker. In this case the proxy for the 3D effect will be the various textures and sizes one can easily generate from the sense of touch.
Great question--summoned my gestalt psych, perception, wundtian elements of consciousness concepts, and sensation and perception memories.
PR Stanley prstanley@ntlworld.com
Sat Jun 7 2008
I think the "testers" have the right idea. Give him half a dozen questions testing his knowledge of the principles instead of this nonsense. An in depth grasp of the principles will help your student work out far more complex concepts on his own than your stairs question. Tactile imagery is not suited to every blind person. Some take to it straightaway, others, in fact a huge number, through no fault of their own find it extremely challenging.
Susan Mooney slemooney@msn.com
Sat Jun 7 2008
This is a concept that actually should have the model of the blocks glued together and handed to the student in such a way that they have the same view as the sighted kid, but since most educational bodies won't consider that, what I have seen done in the past was to have the drawing represented as is, then the side view, the top view and the bottom view.
PR Stanley prstanley@ntlworld.com
Sat Jun 7 2008
The cardboard modeling idea sounds reasonable though.
Susan Mooney slemooney@msn.com
Sat Jun 7 2008
It is reasonable. Unfortunately, our state won't allow it because they say it's an unfair advantage. For some reason they don't consider a flat, tactile representation of a 3-d drawing to be an unfair disadvantage.
Patricia Renfranz dblair2525@msn.com
Sat Jun 7 2008
In the class where this concept is supposedly taught: Is there a way to expose the children to structures like this, even let them build them, so that they have the concept; then introduce tactile graphic (whatever you decide) along with the object with time to study both. Any blocks could be used.
Then, when the student comes across the graphic on the test, they have a chance of understanding the graphic in a real way.
I think it is worth it for a blind student to understand perspective and the geometry of complex shapes, for example, that when one looks at any 3-dimensional object, there is often a lot hidden from sight (or feel, in this case) that contributes to its total volume, or whatever.
I realize my suggestion doesn't really help you with preparing a test. But I do think that for blind kids, some math principles would be much better appreciated if they are given the opportunity to experience the real object or manipulative. Great question!
Patricia Renfranz dblair2525@msn.com
Sat Jun 7 2008
Indeed. 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. Of course, the school didn't provide that person--it was dad. Thanks for your good work,
PR Stanley prstanley@ntlworld.com
Sat Jun 7 2008
So, let me get this right, you're suggesting that a blind pupil should be able to substitute his sense of touch for a pair of eyes. You do realize that the human eye has had a few million years to evolve and develop the ability to process visual data. 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. 3-d imagery works for sighted people because it tricks them into thinking that the 2-d drawing is in fact a 3-d object. 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?
Patricia Renfranz dblair2525@msn.com
Sat Jun 7 2008
A few points:
Yes, my blind child can substitute her sense of touch for a pair of eyes, since she doesn't have any. 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. 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?
I have seen on Math Olympiad problem sets the type of test question being referred to. 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. 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.
PR Stanley prstanley@ntlworld.com
Sat Jun 7 2008
Paul: Forgive me but you seem to be contradicting yourself. Besides, picking on a part of what I had to say is a bit disingenuous. 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.
“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.”
Paul: are you saying that's my opinion too?
<The only time my child is provided objects rather than drawings (often poorly-rendered) and instruction is when her dad or I provide them. 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? I have seen on Math Olympiad problem sets the type of test question being referred to. 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. 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.>
Paul: and perhaps you would have the grace to apologize to me for misrepresenting my argument. Somehow I doubt you have the courage or the decency.
God! Save me from hypocrites! Paul
Susan Mooney slemooney@msn.com
Sat Jun 7 2008
Pat, you're right on. But how many kids actually get that kind of experience? I totally agree that 3-d drawings should be taught in the classroom, but the reality is more often than not the kid has the same (crappy) drawing in his/her textbook and the well-intentioned sighted teacher thinks that because the drawing looks pretty and just like the one in the print book, then the kid's getting the concept. 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.blachman@gmail.com
Sat Jun 7 2008
In connection with the general matter of perspective drawings, I suspect children who want to understand perspective should begin with a picture of a road stretching into the distance, with the sides of the road coming together as the road reaches the horizon line because the angular separation between lines from the viewing position to the two sides decreases as one raises one's gaze from a nearby part of the road to the furthest distance.
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.
Sharon Clark sharonjackson03@comcast.net
Sat Jun 7 2008
Gary,
I have seen other state-wide tests represent this type of question using solid, long dash, short, dash, and dotted lines to represent the different levels.
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.
T. J. tjmaries@yahoo.com
Sat Jun 7 2008
I second Mathew’s suggestion! 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. Mathew’s explanation is very easy for me to understand!
Susan Jolly easjolly@ix.netcom.com
Sat Jun 7 2008
I'm not sure I understand what is the point of this question and why it is on this test. If it is a test of the conventions used to represent three dimensions in a planar drawing then I think it is silly.
If it is a test of how to build a staircase out of blocks without any other materials, then you don't need the drawing. Let's say we want to create a very simple staircase where the rise, run, and width is equal. If you are going to build it out of cubes with no support except a flat surface, then you need three cubes for two steps or six cubes for three steps. For the two-step staircase you put two of the cubes on a flat surface and then put the third cube on top of one of the two cubes.
For the three-step staircase you put three of the cubes on a flat surface for the bottom layer. For the middle layer you put two of the cubes on the bottom layer: one on top of the middle cube and one on top of one of the side cubes. For the top layer you put the last cube on top of the side cube of the middle layer.
Is the point of the drawing for the student to see that this is how the staircase has been built? This bothers me because it is not how a real staircase is built; in a real staircase the stairs are supported on the sides. The space under the staircase is often used for a closet or another purpose. You can build something similar to a real staircase with snap-together blocks like Lego blocks.
Matthew2007 matthew2007@charter.net
Sat Jun 7 2008
This task or test item is probably or simply a test of depth perception. There usually are 1 or 2 test items in math or geometry assessments used to test visio spatio skills and the depth perception I mentioned in a message before. On the other hand, and its probably more than likely the case, this just one of those academic or intellectual bits of wonderment we often experience on our day to day lives such as why man hasn't figured out a way to fly via jet pack or otherwise rather than commute by automobile. this person was probably just asking for some information on a question he is asked or comes across from time to time and innocently thought he would ask the list for information he might not have or couldn't figure out at the moment. I don't think it was something we needed to twist ourselves into pretzels to solve, or to childishly begin insulting others because their limited views were limited to their perception of what is important or not important to assess in what might be a well standardized test of geometric or visio spatio ability. The originator of the question was probably trying to find a way of creating an analogous test question for the blind while trying not to break the endless time and effort that went into creating the original standardized test question for the sighted. In other words, the originator of the question was trying to figure out how he could make a PDA accessible to the blind without writing the software from scratch. He was trying to find a way of giving blind individuals access to off the shelf GPA devices without having to write the code from scratch. He was trying to figure out a way of climbing mount Everest as a blind person. Get the drift those who attacked? I'm not referring to you Susan.
Susan Mooney slemooney@msn.com
Sat Jun 7 2008
Usually the point of these 3-d drawings on end-of-grade and/or end-of-course state-mandated tests is for the student to be able to figure out from a 2-d representation, how many cubes were used to make the drawing. So they have to be able to figure out sides, etc. by visualizing what the bottom, side, back and top views look like. I think it's a volume question so if you just memorize that when you get this kind of question you do LxWxH, I think you'd get the correct answer. However, if it's more than that and asking something like "How many cubes would you see from the side?" I'm not sure (again, not a math prof and don't even play one on TV), it may be more difficult for the blind kid. Not impossible, just a disadvantage. Whether you've got the concept or merely memorized "When I get this, I do that" but don't understand the concept, is a whole 'nutha thing.
The original post was referring to whether or not the proofreader should suggest a different way to represent the drawing which is what proofreaders come up against all the time now since there's so many graphics on the tests. This is a problem because often times you just have someone who makes a nice pretty graphic but it's tactually meaningless or extremely confusing with a lot of clutter. I am always amazed when I meet a competent tactile graphics person because they can--it seems to me--immediately look at the drawing and see how it needs to be represented tactually. I have to sit for days and try and re-try before I finally get it right. Isn't it wonderful that we all have different talents ... but it's difficult finding good tactile graphics people.
I totally agree with Patricia, however, in that if the resource or "vision teacher" isn't teaching these things or if they're simply "teaching the test", then the blind student isn't getting what s/he needs. Luckily, her daughter has the fabulous parents she does because it doesn't sound like her daughter's getting what she needs in school. All too often it falls on the parents and if the kid doesn't have parents with either the abilities to provide that extra educational oomph, the kid's at a big disadvantage. Although you could argue that this happens to sighted kids, too, it usually the norm rather than the exception with blind students ... in my experience.
Although I think we all got a little off topic from the original question, I think it has been an overall decent discussion.
Matthew2007 matthew2007@charter.net
Sat Jun 7 2008
Regarding your statement: "The original post was referring to whether or not the proofreader should suggest a different way to represent the drawing which is what proofreaders come up against all the time now since there's so many graphics on the tests. This is a problem because often times you just have someone who makes a nice pretty graphic but it's tactually meaningless..."
This is the exact discussion I had with a person who makes these raised line drawings for blind students. He showed me a raised line drawing of Winnie he poo standing sideways but turning his head so he would be facing the person looking at the picture. He didn't tell me what this picture conveyed, but since there was a square with a frame on the drawing I deduced it was some sort of picture, I then discovered what appeared to be a couch and television in the picture. Yet I could not figure out what was the image he drew right in the middle of the room. I traced my fingers over what felt like ears, but there weren't any arms or legs so I reasoned it couldn't be any sort of biped or quad (remember Winnie the poo was standing sideways in the drawing." I finally gave up guessing and he told me it was a picture of Winnie the poo standing sideways and facing the person looking at the picture. Once he told me that I ran my fingers across the drawing and sure enough it was Winnie the poo alright, little shirt, big belly and all. He also stated that most of the blind students he had shown the picture to did not identify the figure as Winnie the poo, except for the student who had been blind from birth who identified him immediately. I don't know how that guy was able to identify Winnie the poo never having seen him, and I to this day don't believe he was able to do it, but I'm not excluding any other evidence which I can't come up with right now. For example, maybe that blind student had a type of relief picture of the drawing as a child--who knows--I never asked. My point is that if Winnie the pooh simply been drawn with his arms and legs shown in a complete frontal view, almost all blind individuals would have been able to identify the main character in the picture--that is of course having had prior exposure to who Winnie the pooh. By the way, he was using some sort of new version of the tiger embosser and some additional software. My point? Don't really know as I've forgotten--but its definitely possible to convey many images if one has the desire to think it through and the artistic background necessary to convey depth and so on. I remember as a child going to an amusement part and standing in front of an exhibit where the presenter placed a bottle at the bottom of an incline, he released the bottle, and it somehow rolled upwards--that freaked me out!--sorry but too lazy to lookup the name of this illusion.
Michael Whapples mwhapples@aim.com
Sat Jun 7 2008
I think this question has raised some interesting points, and the answer to the original question really depends on what outcome is wanted (ie. how to allow the question to be answered or how to allow the concepts of 3D visual effects).
If teaching to the test is sought, then Matthew's original idea is good, but it isn't necessarily a "standard" way, I don't know really if there is a "standard" way to do it on a tactile diagram. Even if there was, if the student were to be asked to draw a diagram, then it requires the marker to know how it should be in a tactile form. Also I would say that an explanation sentence or two may be required to go with the diagram so that the student knows what the convention is on the diagram (As Matthew pointed out, it is much easier to understand a diagram once you know what sort of thing you are dealing with).
If an understanding of 3D visual effects is sought (and it can be useful, I will give an example later) then conventions such as weighted shading at best separates your representation from others, and possibly will even hinder you. My example of where 3D visual effects can be useful is in astronomy. There are two (related) examples in astronomy that come to mind:
The explanation of measuring distance to nearby stars due to the fact that the distant stars (which form the background) appear to move less than the near star does as the earth moves around its orbit. The other is estimating the size of objects through the angular separation of its outer edges, and knowing its distance (ie. a small object which is close may appear larger than one further away).
Admittedly those two examples could be explained through other means, but the knowledge of 3D visual effects does aid an initial understanding of generally how things appear, and yes you would still have to go away and calculate it properly in the real world.
Now one last point I want to make. With tactile graphics I separate them in to two categories, diagrams and pictures. By this I mean diagrams are trying to convey information and that it is the information, which is important. Pictures I say are where the appearance is what matters (it normally is what you would visually see if you had an object, just put on to a 2D surface). I would say a "diagram" should always be possible to make accessible, it might not necessarily be a tactile diagram, it might be a description, it might be a bundle of tactile diagrams and descriptions, it might even be a model, it really doesn't matter how it is represented so long as it is efficient for the person who has to read it, and that it gives them an understanding of the information that diagram was trying to give. So in the above example where sighted people understand the idea of distant objects appearing smaller, a more accessible explanation may be to give the student two pyramid with the same base size but of different heights so they can feel how the sides form different angles depending on the height of the pyramid. Pictures I would say generally aren't worth trying to convert, they normally are there for visual appreciation, and may add little, if anything, to the actual understanding of a subject, eg. when I was at university the astronomy module had plenty of pictures of galaxies, but the content of the course was more about how to perform calculations to estimate properties of stars, galaxies, etc. Generally a short description was all that needed to be done for these pictures, if anything was going to be done. Luckily for me at university the person producing the tactile diagrams did know the subject so was able to make intelligent decisions on what to filter out (I suppose that's another point, I should really go back and edit my email to correct where I said I had come to the last point, but there's better things for me to do).
2. Subject: A Regular 10-faced Solid and The Cube
T. J. tjmaries@yahoo.com
Mon Aug 4 2008
Hi everybody, I am looking for these math items. If anyone knows where I can purchase these, information is appreciated!
- Advanced set of Cuisenaire Rods
- Geometric Solids of dodecahedron, hexahedron, icosahedron and octahedron.
Responses:
Jonathan Godfrey a.j.godfrey@massey.ac.nz
Mon Aug 4 2008
If you can't find the solids easily in a large enough size and a small model is sufficient; I would suggest some role playing dice.
The shop would know them as d8, d12 and d20. I'm a little confused by the hexahedron though as the regular solid with six faces is a cube. The d4 is the tetrahedron and the other one often supplied in sets of dice is the d10 and is not a regular solid, having kite shaped faces.
T. J. tjmaries@yahoo.com
Mon Aug 4 2008
Role playing dice? I'm confused. Please explain.
Jonathan Godfrey a.j.godfrey@massey.ac.nz
Mon Aug 4 2008
Hello,
Those of us who played such games as Dungeons and Dragons used various dice to get different results for random aspects of the game.
D&D and a very large number of games like it were collectively known as role playing games (RPG)
The RPG market is different today but still survives, albeit on a much smaller scale than in the past.
From the six different dice, d4, d6, d8, d10, d12 and d20, we could also generate results that would mimic a d2, d3 or d100 results. I learnt a great deal more about probability from RPG than my first year course in Statistics. I well remember arguments about the virtues of rolling 2d4 (the sum of two four-sided dice) over 1d8 (single roll on eight sided dice) even going as far as seriously thinking of the variation of each vs. the expected value. I still have the dice actually even though I haven't been able to read the print on them for over 20 years.
T. J. tjmaries@yahoo.com
Mon Aug 4 2008
Ok! I think I understand now. So, if I do understand correctly role playing dice is any 6 sided dice right.
Jonathan Godfrey a.j.godfrey@massey.ac.nz
Mon Aug 4 2008
No, the dice have different numbers of faces. 4 for a tetrahedron, 6 of the cube, 8 for the octahedron, 10 for a nonregular solid, 12 for the dodecahedron and 20 for the icosahedron. There is nothing special about the cube or six sided dice that would make them more useful for role playing although I do know of at least two RPG that relied on them alone.
tribble lauraeaves@yahoo.com
Tue Aug 5 2008
I'm curious -- what is the shape of the faces on the 10 sided die? Is itcalled a nonregular solid because the sides have different shapes, unlike the other dice in your list? Curious.
Jonathan Godfrey a.j.godfrey@massey.ac.nz
Tue Aug 5 2008
Hi,
There are the five regular solids that have equal faces and equal angles between faces and therefore all pairs of adjacent edges.
While it might be possible to create a ten faced solid with equilateral triangles, the appearance of the solid would change depending on which face was towards you. This is best illustrated by the point that it has only one axis of rotation. A cube has many axes of rotation, being through all opposing faces, opposing edges, and all opposing corner points. Note this ignores axes of rotational symmetry that are based on the trivial 360 degree rotation.
I've got an idea the regular solids can also be called Euclidean solids but we're going back a few too many years to my axiomatic geometry course to be utterly sure.
Nelson Blachman nelson.blachman@gmail.com
Tue Aug 5 2008
If you write the numbers from 1 to 10 on half the faces of a regular icosahedron and then the same numbers on the other 10 faces, you'll have a regular "10-faced die." It will be "fair" because all ten numbers will have the same probability, 0.1, of turning up as opposed to unequal probabilities in the case of an irregular decahedral die.
tribble lauraeaves@yahoo.com
Wed Aug 6 2008
Hi Nelson --
Well, using that logic, you could call a cube a 3 sided die by duplicating the numbers on half the faces... and similar numbering could transform other die as well... But that's a stretch. Interesting...
Jonathan Godfrey a.j.godfrey@massey.ac.nz
Wed Aug 6 2008
We did that exact thing to get two sided dice and three sided dice. And your other question was on the use of triangles to get any even numbered dice. Possible but eventually you find that the isosceles triangles get very acute angles on them. I would think a 14 faced dice might be possible, but going too much further would get a wobbly dice. By the way, no one has asked about how one reads the four faced dice.. The number was written on the apex angle of every face. It meant that there were three numbers on each face. I think this is the last message I'll write on this topic. I reckon we've done the thread to death.
Nelson Blachman nelson.blachman@gmail.com
Wed Aug 6 2008
I agree that we've had enough of dice (which is plural, the singular being die). Let's just use a bunch of distinct coins instead. For the penny, heads will represent 1 and tails 0. For the nickel 2 and 0, for the dime 4 and 0, for the quarter 8 and 0, and for the half dollar 16 and 0.(For those outside North America, try a farthing, a halfpenny, a penny, athree pence, a sixpence, and a shilling.)
Don't use any more coins than are needed for a total of the largest number you want minus 1.
Then toss the coins, sum the values of the heads, and add 1 to give you the fairly chosen random number you want. If this total exceeds the largest value you permit, ignore it and toss again.
Nelson Blachman nelson.blachman@gmail.com
Tue Aug 5 2008
TJ,
The usual dice have six faces each, not six sides; a hexagon has six sides. A single die has six faces.
A cube is the three-dimensional analogue of a square, and its four-dimensional analogue is called a tesseract. It has eight cubic cells,24 square faces, 32 edges and 16 vertices. It might be worthwhile to consider RPG in four dimensions, then five, etc
Maybe I should have mentioned that the cube (also known as the regular hexahedron) has not only six square faces but also twelve edges and eight vertices before going on to describe the tesseract. When a tesseract is rolled across a four-dimensional table (with a flat three-dimensional surface called a hyperplane), one of the eight cubic cells will end up in contact with the table's hypersurface, while six other cubic cells will rise from the table top and join the eighth cubic cell, which will sit above the one in contact with the table and parallel to it, just as thee top surface of the cube is parallel to the one in contact with the table in three dimensions after the cube (or die) stops rolling.
T. J. tjmaried@yahoo.com
Tue Aug 5 2008
OK. I understand it except for the words "Euclidean" and "axiomatic".
Nelson Blachman nelson.blachman@gmail.com
Tue Aug 5 2008
T. J.
"Euclidean" is the adjectival form of "Euclid," the ancient Greek who derived the principles of geometry from a set of assumptions called axioms." He proved that there are just five regular solids, not realizing that, by extending the faces of the regular dodecahedron and the regulari cosahedron, he could obtain four more regular solids whose faces intersect each other,while in the first five the faces do not intersect each other, and those five solids are convex, i.e., the line segment joining any two points on the surface lies entirely inside the solid.