Wikipedia:Reference desk/Archives/Mathematics/2008 May 12
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- 1 May 12
- 1.1 Odds of winning in Lotto 6/49
- 1.2 Re: Trigonometric functions
- 1.3 Mathematics-Quadratic Equations
- 1.4 Left/right division
- 1.5 Connection between the definition of the unit sphere and the real-projective plane
- 1.6 Non-degenerate conic in the real-projective plane
- 1.7 Combinatorial problem
- 1.8 nets for making geometric solids
Odds of winning in Lotto 6/49
Per 6/49, the chances of winning the jackpot is roughly 1 in 14 million. However, lotto 6/49 consists of matching 6 numbers from a box of 49 numbers. So therefore, shouldn't the odds of winning the jackpot be 1 in (69 x 68 x 67 x 66 x 65 x 64) or 1 in 86311779840? Where have I gone wrong? Acceptable (talk) 00:50, 12 May 2008 (UTC)
- First, you morphed 49 into 69 for no apparent reason. Then you failed to make the distinction between combinations and permutations. There are indeed 49*48*47*46*45*44=10068347520 different possible drawings, but they're not all considered distinct from each other. If your ticket is "1,2,3,4,5,6", you will win whether the drawing is "1,2,3,4,5,6" or "6,5,4,3,2,1" or "3,5,6,2,4,1"... There are 6*5*4*3*2*1=720 such orderings, so you will win 720 out of 10068347520 drawings instead of just 1 out of 10068347520. The number in the article, 13983816, is 10068347520/720. —Preceding unsigned comment added by Tcsetattr (talk • contribs) 02:57, 12 May 2008 (UTC)
- (Edit conflict) Your approach would be correct if you needed to have the numbers in the same order, but that is not the case.
- Take a simpler example, choose 3 numbers from a total of 10, suppose the winning combination is 1 2 3, then you have in total 3! = 6 ways of choosing the combination 1 2 3, not just only one, which means the odds are 1 in (10 x 9 x 8)/6 = 1 in 120, instead of 1 in 720 if the order mattered and 1 2 3 was different from 2 3 1.
- Similarly, for the 6/49 lotto, the odds are , where ! is the factorial sign.
- So the formula is the same as yours except it accounts for the different ways you have of getting the winning combination, as .
- Hope that helps. -- Xedi (talk) 03:08, 12 May 2008 (UTC)
Re: Trigonometric functions
The article at http://en.wikipedia.org/wiki/Trigonometric_function states, "Trigonometric functions are commonly defined as ratios of two sides of a right triangle containing the angle, and can equivalently be defined as the lengths of various line segments from a unit circle. More modern definitions express them as infinite series or as solutions of certain differential equations, allowing their extension to arbitrary positive and negative values and even to complex numbers."
1. The example of the triangle shown to the right side is NOT the right triangle which is the trigonometric triangle. The triangle shown linearly (and incorrectly) demonstrates ratios only for the sin, sec and tan and displays them all wrong. That it is not the trigonometric triangle is evident in that a SECOND illustration is necessary to linearly display those ratios for the cos, cot and csc as linear expression of functions. NOT ONE TRIGONOMETRIC TRIANGLE, RATHER TWO ERRORS.
2. Modern definitions, and therefore the rote of modern mathematics, fail to recognize two very interesting and distinct functions which are quite present in the trigonometric triangle. Those functions are the dav and codav. They are so present in the trigonometric triangle they cannot be denied. One just needs to first achieve an understanding of the Trigonometric Triangle.
To do so see Google Videos
Search term: trigonometric triangle trilogy
or view (to make it easy) at these Google links:
(self-aggrandizing spam links removed)
- Yes, clearly, we are all going to change our understanding of mathematics, which has existed for centuries, because of YouTube videos uploaded by you. Please do not troll the reference desk. --Kinu t/c 02:29, 12 May 2008 (UTC)
- Dear god, that was boring. And disappointing. I'd been hoping for some insanity-fueled entertainment, maybe a rant about timecube or something, but it's not. It's just describing a way to fit all six of the ordinary trig functions into a single diagram as the lengths of line segments. BTW, a XZ is a function of the angle. The angle isn't a function of XZ. Black Carrot (talk) 08:36, 12 May 2008 (UTC)
- C'mon guys, that was a bit harsh. I don't think he deserved to have his head bitten off for attempting to discuss pedagogical techniques for trigonometry. Assume good faith, even here on the refdesks. Maelin (Talk | Contribs) 14:24, 12 May 2008 (UTC)
- Perhaps you are right in that the gun was jumped, but it is easy to assume shenanigans when someone comes in with some links to videos they created, talks about a function named after themselves, and proceeds to imply that the foundations of trigonometry are all wrong. Perhaps had the poster worded things a little more... diplomatically... i.e., asking for feedback on his methods rather than pointing out "errors" in our ways, it would yield to providing more constructive commentary. --Kinu t/c 22:50, 12 May 2008 (UTC)
- C'mon guys, that was a bit harsh. I don't think he deserved to have his head bitten off for attempting to discuss pedagogical techniques for trigonometry. Assume good faith, even here on the refdesks. Maelin (Talk | Contribs) 14:24, 12 May 2008 (UTC)
- Having now viewed the video in its entirety, I have a couple of comments. Firstly, don't feel too discouraged by the above responses. Any self-motivated explorations of mathematics are a worthy goal. Now, a few comments about the actual video
- Your understanding of what constitutes a mathematical "function" appears a little off - you seem to require that functions be "unique" over the domain, hence your dismissal of the function yielding the length of XY since it was equal for complementary angles. This property, called injectivity, is an often-desirable but by no means essential property of functions. Remember, even sine and cosine aren't unique if you consider their domain to be all 360° of a full circle.
- Whilst your construction of the triangle certainly yields valid definitions of the trigonometric functions, it does not do so in a particularly motivating sense. The construction seems somewhat arbitrary and contrived. Teaching school students trigonometry is difficult enough, and that's when the function definitions lend themselves intuitively to simple applications. A construction like this one would serve as merely an extra layer of complexity in an already challenging new area for young students.
- Your claims that mathematics has "failed to recognise" the functions yielding the lengths of WZ, VX and XY are somewhat unfounded. They are easy to define, as given the angle θ, we can prove that WZ = OZ - OW = sec(θ) - cos(θ), that VX = 0X - OV = cosec(θ) - sin(θ), and that XZ = XY + YZ = cotan(θ) + tan(θ). What mathematics (or, more accurately, mathematicians) hasn't done is give these functions special names, probably because they are not especially useful or interesting.
- Trigonometry offers many opportunities for "Ooh, neat!" moments where you discover that various identities can be proven geometrically. Good luck in your trigonometric adventures! Maelin (Talk | Contribs) 15:10, 12 May 2008 (UTC)
- You know, after looking at the history of the article, I now fully understand the first part of Mr. Davie's comment... which brings up an interesting point: why was this image done away with for the current two-graph system of defining the ratios? (Perhaps this is a discussion best suited for the article's talk page, but I thought I'd pose it here for starters.) --Kinu t/c 00:08, 13 May 2008 (UTC)
- After having read most of the above, davie, your ideas are certainly worthy of discussion but your presentation is apt to offend those with significant mathematical knowledge. I to have had to learn how to pose my thoughts in a manner that avoids me coming off like I am the god of math or what ever. I find the best tool is to pose one's arguments a question (after all this desk is for questions not rants). As for those who quickly concluded that his post was trolling, I think the lesson here is don't assume something is true just it fits the bill, look a little deeper into whether or not his arguments make any sense, if not then such a post would be motivated, but if they do seem to have some merit, I would say one should address it as any other question. A math-wiki (talk) 06:15, 13 May 2008 (UTC)
Amusing. Videos posted on YouTube.
"your dismissal of the function yielding the length of XY"
Prejudice is an ugly bedmate.
The functions which above are described as simple subtractions and additions in the linear geometric perform exponentially when studied through 91 degrees (0 through 90), hence sin(sq)/cos and cos(sq)/sin are more appropriate descriptions.
Regarding an 'extra layer of complexity", there is far more elegance in the Trigonometric Triangle of my video than either two triangles or a first quadrant angle with a triangle in the fourth quadrant, both products of Wiki. Imagine using one triangle to measure the angle. Oh my, too complex for Wiki.
And if one starts counting at angle no. 1, like the trig tables tell us, at zero degrees (it is not to be feared), when one arrives at 360 degrees that is angle 361. Good grief, I am far to radical to be taken seriously, claiming there are 361 degrees in a complete circle. Have a fun time with that at my expense guys. :)
- I defend you when others assume that you are just another Timecube madman, and offer you constructive criticism about your ideas and try to help correct your misunderstandings about mathematical conventions, and you mock me? Your approach to mathematical communication with others needs some real work. Maelin (Talk | Contribs) 02:09, 14 May 2008 (UTC)
- You have confused the fenceposts with the gaps between them. Given 90 gaps (the first degree, the second degree, and so on up to the 90th degree), there are 91 fenceposts (at 0, 1, 2, ..., 88, 89, 90 whole degrees). So what? I've planted 91 fenceposts down and gotten 90 gaps. The gaps are the degrees. When I go to the next quadrant, I need to add 90 fenceposts to add 90 more gap (because I already have a fencepost labeled "90"). Continuing around, I would plant a 361st fencepost, but I don't need to -- I've already got one there labeled "0". So I would plant 361 posts to get 260 gaps, but because I've curved my fence around, I can get the first post to be the last post and therefore get 360 gaps between 360 posts.
- This is no stranger than realizing you have to go one inch along a ruler to move between two inch labels. The gap between the labels is the inch. The angle between the degree labels is one degree. -- Fuzzyeric (talk) 02:25, 18 May 2008 (UTC)
Excuse me, Maelin, if you took offense, but I do not recall that I dismissed a "function yielding the length XY". I do appreciate the kind words you have offered. Perhaps you were referring to that which Black Carrot commented upon, and you made a typo intending XZ and not XY. And I forgive Black Carrot, who refuses to recognize these two functions and finds the Trigonometric Triangle only good for a way to fit all six of the ordinary trig functions into a single diagram. It's eight functions which fit in, with the versine and the exsec. as bonuses. Yeah, all the functions are there in one triangle in the first quadrant.
And while my understanding of what is a function may be somewhat wanting, I cannot find where I said, as Black Carrot seems to allege, that the angle is a function of XZ. My point is that the Trigonometric Triangle, i.e., the triangle which measures the functions, is constructed from the given angle in terms of (or perhaps in relation to) the unit circle.
And while the functions may repeat in the second, third and fourth quadrants, that could, with an added layer of complexity, be resolved by a functional notation for each quadrant. Each angle is unique in its input and an angle of 173 degrees does not occur in the first quadrant. Perhaps math has a notational issue to deal with. Oh my.
As to the argument, What mathematics (or, more accurately, mathematicians) hasn't done is give these functions special names, probably because they are not especially useful or interesting, it may also be that, given the triangles mathematicians have used to measure the angle have not been a triangle which encountered these distinct functions. Given the examples of triangles mathematicians have used to measure the angle, I believe there is a stronger case to be made for the triangles mathematicians have used to measure the angle have not been a triangle which encountered these distinct functions.
While it may appear that I have 'come off as a god of mathematics', I would suggest that the argument 'because they are not especially useful or interesting' raises the questions of to whom and isn't that denial of openness of knowledge itself acting as a god of mathematics. Obviously, if I am the pot then I am good company with the kettles.
It ain't about me folks. It's about elegance of solution and knowledge. While the exercise here is somewhat enjoyable, the videos are there and they will gain viewers. And as they do they will generate further discussion. And I am certain that in time the Trigonometric Triangle and the two functions that math has shunned for whatever reason will, of necessity, become accepted.
As to the added layer of complexity, I take it that concern is in regards to teaching minds which are less than inquiring and not in regards the communication of knowledge, which is what separates homo sapiens from all the other species.
- Davie, you seem with that last post to be mostly disregarding the time tested advice I gave in how you should present your arguments, I do find these functions of interest. It is an unfortunate consequence of human nature that one's interpersonal skills will significantly affect the reception of one's ideas, by most listeners. If I am correctly interpreting what those above have posted, they seem to me to be saying the reason those functions are seldom discussed is that they can be described by the 6 functions which are typically described in math class. I do still thing discussing them is a worth use of one's time, but in order to even do so, one should present their arguments in a way that successfully communicates the argument as well as respects the fact that in this case the audience is mostly people with significant knowledge of the general subject being discussed. As for the 361 angles in a circle, first there are an infinite number of them, since an angle need not have an integer measure in degrees, but for those angles that do, there are only 360 distinct angles, this can be shown by observing that the angle of 360 degrees corresponds to the point on the unit circle that the angle of 0 degrees does as well, and thus angle no.1 as you call it, and angle no. 361 are two different names for the same angle. A math-wiki (talk) 07:00, 14 May 2008 (UTC)
Yes, thank you for your advice. While it can appear that zero degrees and 360 degrees are the same angle, that is deception of appearance. Is the angle of 360 degrees a fourth quadrant angle? I know that the angle of zero degrees is where I am when I put the compass point and lead to paper. As to the two functions can be described by the 6 functions, I observe of the eight functions, six functions can be described by the quotients sin and cos. Let us say that such descriptions are theoretical. Similarly the eight functions can be described numerically. Similarly the eight functions can be described geometrically. What parts of that knowledge should we dismiss? Is it possible the rote typically described in math class is stifling inquiring minds? I don't know. I haven't been in a trig class in nearly half a century and I don't feel like my mind is stifled.
As to the dismissal of XZ as a function (mentioned above) I said, However, such a conclusion regarding XZ is an error. Perhaps the first clue is that XZ has no cofunction. As we observed earlier, the sine of one given angle is the cosine of that given angle's complimentary angle, which allows for the input of each angle to result in a unique output for the function. However, as can be seen in this image, complimentary angles have the same value for XZ, which lack of uniqueness precludes XZ from being a function.
The image shows complementary angles 27 - 32 and 63 - 58 respectively sharing the same values for XZ, Aside from the special case 45 degree angle which is its own complementary angle, other acute angles and their complementary angles share no value for the same function, keeping it simple here with first quadrant acute angles. For instance the values of the eight functions of 27 degrees is unique to those functions of 27 degrees. What would cause an exception for XZ as a function of a unique angle to provide output which is not unique?
Please keep in mind regarding my initial query that while I took issue with the triangles which Wiki has chosen to display, I offered a solution. —Keith Davie (talk) 13:43, 14 May 2008 (UTC)Keith Davie
Regarding the comment about 360 being the same as 0 degrees. Like I said, it does give that appearance but the appearance is deceptive. I than asked, isn't that a fourth quadrant angle.
Dance of the Trigonometric Tables
I awoke this morning thinking of the dance which occurs between the cotangent and tangent as an angle increases from 0 to 90 degrees, at least in the model ballroom of the Trigonometric Triangle which I provide. At 0 degrees the tangent's value is zero and the cotangent dances away from the tangent through the first quadrant to the eternity of the plane. As the angle progression occurs the tangent lengthens and the cotangent shortens. The cotangent and tangent are always in contact on the circumference at the point where the rising side of the angel, the radius, intersects. Finally the cotnagent arrives at the spot on the dance floor where its value is zero. Here the tangent dances away from the cotangent through the first quadrant to the eternity of the plane.
In my ballroom's model, when the dance continues into the second quadrant with the first hint of obtuseness, the cotangent begins to appear and the tangent performs a grand jeté into the second quadrant. And so their dance continues with the cotangent lengthening and the tangent shortening to 180 degrees where the tangent again is zero and the cotangent dances away from the tangent across the dancefloor of the second quadrant to the eternity of the plane.
And this dance repeats again into the third quadrant where the cotangent performs a grand geté and then into the fourth with the tangent again performing the grand jeté as it enters. And when the couple arrive at degree 359, the 360th angle of their Dance of the Trigonometric Tables, they have not completed the dance. There is one more angle before the dance is where it began. And when they take that 361st step the tangent is at zero but the cotangent is still in the fourth quadrant, dancing across its dancefloor to the eternity of the plane.
So while the appearance of the angle may look the same, the Trigonometric Triangle reveals a different image and the deception of appearance.
- 0 degrees and 360 degrees are neither first nor fourth quadrant angles, they are a quadrantile (or a border of two quadrants). If one starts on angle number 1 as you say, then their first step is to angle #2, and so forth, so their 360th step is to and angle #361, which returns them to angle #1 A math-wiki (talk) 19:50, 15 May 2008 (UTC)
Thank you for your responses. I have reviewed them with a bit of distance from the jostlings of the moment.
I see where one could take offense at my original bold statements.
Especially rankling seems to have been my statement that modern mathematics fails to recognize these two functions. Perhaps I could have been more gentle, but consider the two triangles currently displayed at Wiki. Neither mentions nor shows their existence. That is hardly an acknowledgment.
Similarly the first degree angle image which Kinu mentions does not mention these two functions, rather it shows the versine and exsecant, i.e., two functions where one will do. An odd complexity in my opinion. Especially since neither is defined by the sin and cos as the other functions, including Davie's. That image also has the problem of an unnecessary and less inelegant extension into the fourth quadrant. Where's the acknowledgment in that triangle?
Trig Tables? No acknowledgment. Nor did anyone post a reference which showed that the Trigonometric Triangle and the two functions I claim as Davie's functions had been considered and rejected by peer review.
And while T'hey are easy to define, they haven't been until now, hence Davie's Functions. It is also easy to mis-define them. It's like those tools do not exist for mathematicians. It should not be surprising then that the tools are not used.
As to the other issues raised by the responders, many were either errors such as typos and incorrect references or flat out misrepresentations of what was said by me in the videos and even on this board.
Concerning my dismissal of XZ as a function, I understand full well the argument that arises with 2nd, 3rd, and 4th quadrant sharing of values. If you watched all three of the videos you know that I addressed that issue as the usual considerations. If we don't make those considerations, then math has a notational issue. We can't have these things both ways.
In regards to first quadrant angles, the acute angles with which I was dealing, I am accurate in my statement and my mathematics apparently since no one has yet answered my responding query, What would cause an exception for XZ as a function of a unique angle to provide output which is not unique?. To that I will add, where is it's co-function, and if it does not have one, what gives it that exception also?
The final thing which everyone seems to have missed is in the very first sentence throughout my video trilogy I say, ...I am not a mathematician. Instead of taking me to task for my inelegant verbal and written expressions perhaps one of you could have seen the opportunity which these tools presents to a mathematician to more elegantly and fully discuss these tools (Trigonometric Triangle model and Davie's Functions) than that which I am capable or trained to do. Good tools belong in the mathematics toolbox so their uses can be found or realized by others. What if Fibonacci had been poo-pooed because there was not Dow Jones so he could not have known his theory would be applied to that market by someone inspired by his work? If one is denied the tools, one will never use them.
I have considered the things which I put in my videos for over thirty years and am confident in the math and speak confidently of the Trigonometric Triangle model I presented and Davie's Functions, which are best defined through their theoretical exponential expression of sin^2/cos and cos^2/sin, for that is how they act in linear, geometric progressions over several angles. My model is based in part on the standard math tables and is in agreement with that.
I now have posted one more video at Google Video. Those interested will look. I am not posting the link so Wiki Math Reference Desk will not need delete it.
This is not about self-aggrandizement but about the dissemination of knowledge.
Now playing at Google is a silent movie called Function Cotillion (sub: Having A Ball With Geometry), presenting a set of dances displaying the actions of the eight functions as dance partners, geometrically expressed in the first quadrant.
It too might provide someone some inspiration some day.
I made an offering of a model when I came here. It has been rejected because it is boring. It appears a high point in mathematics has been achieved at Wiki.
Again, thank you all for your responses. Even the negatives have made me more confident in the mathematics and elegance of my model. Nothing better has been offered.
I have no regrets.
I regret to inform that the video I first uploaded to Google is not playing correctly, dropping a lot of data, nor did it download correctly. I have just uploaded a replacement, but it may take a while for the upload to take effect. I will check later. If that one does not work, I will try a very low-grade and blurry format as an upload. I will check my talk box later and if anyone is interested I will attempt to email you a copy if you tell me where. It runs in less than three minutes and is less than one meg.Keith Davie (talk) 22:50, 16 May 2008 (UTC)Keith Davie
- Keith, you seem to be missing the point somewhat. We don't bother naming and defining and teaching high school students the "davie" functions simply because they are not useful. Your claim that they are "missing from the mathematics toolbox" are somewhat akin to saying that plumbers' toolboxes are missing wrenches with flashlights stuck on the side. Sure, you can imagine contrived but possible situations where they might be useful. But they wouldn't be so often useful that hardware companies would manufacture them en masse. Any plumber who actually finds himself needing one would simply make it himself, sticking his normal flashlight to his normal wrench with duct tape. The sine, cosine, and tangent functions are all present in a mathematician's "toolbox". So are the operations of multiplication and addition. Since the davie's functions are a) easily constructible from the other functions, and b) not really useful in regular applications, mathematicians have not seen the need to give them a unique name. You are welcome to name them whatever you wish in mathematical documents and works that you produce (so long as you define them) but it is pure fantasy to demand mathematicians universally name some elementary and seldom useful functions after you simply because nobody else has named them formally yet. Maelin (Talk | Contribs) 13:35, 18 May 2008 (UTC)
I am pleased to inform that the movie Function Cotillion has been uploaded successfully at last at Google Video. The QuickTime movie, which runs in just under three minutes, downloads adequately as well. I had to enlarge the video to ~3 mb.
Maelin, I have answered your at my talk.
The solutions to a particular equation can consist of different kinds of real numbers,for example rational numbers,or irrational;they can be equal roots or unequal,positive,negative or zero.There can be even no real solutions at all.All of this depends on the values of a,b and c in a given equation. The quantity under the square root(delta) controls what kind of roots there are for a given equation.For example if this quantity is negative then there are no rea roots.
QUESTION- What are the conditions that apply to delta, so that the solutions will fit into these categories-real,unreal,rational,irrational,aqual,unequal? —Preceding unsigned comment added by 220.127.116.11 (talk) 14:23, 12 May 2008 (UTC)
- The quantity you call delta is the discriminant of the polynomial. That articles answers all your questions, except the question of (given that the roots are real) when the roots are rational. Assuming the coefficients of the polynomial are rational, the roots are also rational iff the discriminat is the square of some rational number. Algebraist 14:27, 12 May 2008 (UTC)
This is a question about notation. Specifically I’m working in a noncommutative ring with nonequivalent left and right division algorithms. The question is if there is any accepted notation for saying that “a left divides c” or “b right divides c”. Clearly, “a|c” won’t work because it’s ambiguous if a doesn’t both left and right divide c. GromXXVII (talk) 15:15, 12 May 2008 (UTC)
- I don't know if there's a standard notation or not, but if I had to make one up on the spot, I'd go with and . --Tango (talk) 15:47, 12 May 2008 (UTC)
- See quasigroup, or the matlab documentation for a reasonable use of / and \. JackSchmidt (talk) 05:24, 13 May 2008 (UTC)
- If you just mean a if a left multiple of b, then just say a in Rb. One sided ideals are made for this. JackSchmidt (talk) 05:25, 13 May 2008 (UTC)
Connection between the definition of the unit sphere and the real-projective plane
What is the connection between the definitions of the unit sphere and the real-projective plane? I was told they were connected. I have seen how the geodesics on the sphere correspond to lines in the real-projective plane, but when you just look at definitions of spaces I get all confused.. Also, how is it that the real-projective plane can be regarded as a disc? I can't find an explanation of this anywhere! —Preceding unsigned comment added by 18.104.22.168 (talk) 15:26, 12 May 2008 (UTC)
Perhaps Stereographic projection is what you are looking for? The real projective plane cannot be regarded as a disc unless you puncture it.-- Meni Rosenfeld (talk) 15:56, 12 May 2008 (UTC)
- I have an exam in projective geometry in a little under 3 weeks, and I'm not all the confident about passing it, so take this with a large pinch of salt, but, as I understand it, the complex projective line is homeomorphic to the sphere, but the real projective plane isn't (C and R2 are homeomorphic, but when you go to the projective versions, they're no longer the same - C just gets a single point at infinity, R2 gets a whole projective line at infinity. It also isn't a disc... If memory serves, the real projective plane is a sphere with a disc removed and replaced by a Moebius band. --Tango (talk) 15:55, 12 May 2008 (UTC)
- Perhaps you will find Real projective plane useful. -- Meni Rosenfeld (talk) 15:59, 12 May 2008 (UTC)
- (after edit conflict) The real projective plane is the space of lines in R3 that pass through the origin. Since each of these lines intersects the unit sphere in just two antipodal points, the real projective plane has the same toplogy as a unit sphere in which opposite pairs of points are counted as identical i.e. the quotient of the unit sphere by the antipodal map. You can also think of the real project plane as being the compactification of the real plane R2 formed by adding a "projective line at infinity" to R2 - think about how the lines through the origin in R3 intersect with the plane z=1.
- Don't confuse this construction with the one-point compactification of R2, the "complex projective line" or Riemann sphere, which is topologically equivalent to the unit sphere without identification of opposite points. Gandalf61 (talk) 16:02, 12 May 2008 (UTC)
- One can "think of" the real projective plane as a disk, but with each point on the boundary of the disk identified with the boundary point opposite from it (i.e. identify antipodal boundary points). The way to see this is to think of RP^2 as the unit sphere with antipodal points identified, as Gandalf61 discusses, and then throw out the (open) bottom hemisphere, leaving only the (closed) top hemisphere. This is fine because each point in the (open) bottom hemisphere has an antipodal point in the (closed) top hemisphere. When we identify antipodal points to get RP^2, what we get is the top hemisphere (which is topologically a disk), but with antipodal boundary points identified. kfgauss (talk) 20:21, 12 May 2008 (UTC)
Non-degenerate conic in the real-projective plane
I have exams coming up in two weeks and this is a past paper question, i'm worried something similar will come up again this year.. can someone please explain? If you had 5 points in the real-projective plane, no three of which are collinear, how would you prove that there is a unique non-degenerate conic passing through each of the points? 22.214.171.124 (talk) 15:40, 12 May 2008 (UTC)
- Here's an outline of one way to prove this:
- 1) Show that you can make a (linear) change of variables so that three of your points are (1,0,0), (0,1,0), and (0,0,1).
- 2) Write down the general equation for a conic, ax^2 + by^2 + cz^2 + dxy + exz + fyz = 0, and impose the condition that the above three points solve your equation. Write down the general equation for a conic through those three points.
- 3) Show that, given two more points A and B such that no three of these two and the above three are collinear, (a) none of the three coordinates of A or B are zero (so we may normalize z to be 1 and work in the plane with these two points), (b) if we do normalize z to be 1 and work in the plane, A and B are not collinear with the origin. (One can draw more conclusions, but we won't need them.)
- 4) Working in the plane, let A = (x1,y1) and B = (x2,y2) and write down the two equations for these two to be solutions of the equation you found in (2). Show that (up to multiplying the coefficients by a scalar) there is a unique choice of a through e (from (2)) such that A, B, and the three points from (1) are solutions. You'll need to use what we learned in (3). Hint: If you know some linear algebra, this step shouldn't be messy at all. Try to turn it into a question about showing that the kernel of a certain matrix is 1-dimensional.
- This solves the problem, as the conic we find is necessarily nondegenerate. For completeness, you may wish to argue this, i.e. 5) Given 5 points on a degenerate conic, at least 3 are collinear. Hope this has helped. kfgauss (talk) 20:05, 12 May 2008 (UTC)
I am trying to figure out how many permutations I can have of 4 values, where the values are integers from 0 to 100 inclusive. The four values sum to 100 and the position is important. I have figured out that if I have 1, 2, or 3 values, the answer is 1, 101, and 5050 (101 choose 2). How can I do the next step for the 4 value case? I figure this is a nested binomial coeffienct or choose 2 problem. (I realize a good upper bound is 101^4 ~ 10^8. I have calculated 2.5 x 10^5 as a close solution, but I'd like to be exact if possible.)
Also, I am thinking about a case with a larger number like 23. Where I have 23 integer values that sum to 100 inclusive. This would be a massive nested problem. Thanks for your help. --Rajah (talk) 22:22, 12 May 2008 (UTC)
- You are looking for Stars and bars (probability), in particular theorem 2, with and . The answer is . The sequence goes 1, 101, 5151 (not 5050 like you wrote), 176851, 4598126,..., and for 23 you have 941388310113310598422026. -- Meni Rosenfeld (talk) 22:40, 12 May 2008 (UTC)
nets for making geometric solids
Hi. These are not specific homework questions that I have to figure out, nor are they things I have to figure out by myself using some kind of formula, as I am asking for a way to measure these and not the answer to any questions or specific measurements (and am seeing if my formulas are correct). These questions are about making nets for constructing a shell of a geometric solid. First question I have is about the net of a right circular cone. I am to make a net so that the edges so not overlap and to only use tabs for connection. The net for a cone consists of a circle and a shape that is almost like a geograhpic isoceles triangle, where the congruent edges are straight but the edge connecting to the circle is an arc. I understand that the arc's length is equal to the circumfrence of the circle, but how do I draw the arc to the right curvature? I have the height of the cone as well as the diametre of the circle. If you extend the arc, what shape is it and how large is the resulting shape? I'm thinking that it will be a circle, with the centre of the circle at the vertex of the cone's net, as the distance from the vertex to the bottom of the shape which connects to the circle should remain the same, as this is a right cone and the distance on the curved face of the cone between the vertex and the circular edge should be the same. Am I correct, or is there a different simple formula for this? Thus, is the formula for the distances between the vertex and the round edge on the triangle-like portion of the cone's net [math] sqrt(r2+h2) [/math]? Sorry, I'm not very good with LaTeX syntax and r means the radius of the circle and h means the height of the cone. Ok, next I am to do a chevron-based pyramid. What I mean by that is, the base is a chevron shape, which is kind of like, in this case, an isoceles triangle with another triangle cut out of it at the only edge not congruent to the other two edges, that is also isoceles. So, the end result (chevron) should be a quadrilateral with three acute angles, and one reflex angle, and the reflex angle, were is subtracted from 360 to make an acute angle, would be greater than the angle that is pointing in the same direction and is not the two congruent acute angles. Basicly, I hope you know what a chevron is. The vertex for the pyramid is directly above the reflex angle, and the resulting pyramid should have 5 faces, 8 edges, and 5 verticies. However, that's not my question. In order to make a net, will I need to use any trigonometric functions (sine, cosine, tangent) to calculate the lengths of the edges for each of the 5 faces for the net, other than the basic Pathegorean theorem? I know the distance between the two congruent angles on the chevron (the width), the length between the other acute angle on the chevron and the reflex angle on the chevron (the length1), the distance between the reflex angle and the middle of the width line, which forms the edge of the nonexistant triangle in the chevron that does not share the edges with the real shape (the length2) [length1 and length2 should form a straight line with each other, as well as the height, which is the distance between the reflex angle and the top vertex of the pyramid. My other question is, is there a way to calculate whether or not a net with 6 congruent squares will succesfully form a cube if folded, without actually folding it? Something like x edges to seal, x edges to fold, etc. Thanks. ~AH1(TCU) 22:34, 12 May 2008 (UTC)
- Right-Angled Cone: You're right, it'll be a circle for the base and a piece of a circle for the rest, and the radius of the partial circle is equal to the distance from the vertex to an edge of the base, sqrt(r2+h2).
- Chevron: You can do it all with the Pythagorean theorem. Just draw a picture of the base chevron, including the width and height lines, and start filling in numbers until you've labeled every line. Once you've got the base, three of the vertical edges are hypoteneuses of right triangles, with the fourth being their shared side.
- Cube: The number of edges to fold is always five, and the number to seal is seven, and the perimeter is therefore fourteen. That doesn't narrow it down enough, though. Try this. There can be no more than four squares in a row, and there should never be a c-shape anywhere in it. Black Carrot (talk) 02:13, 13 May 2008 (UTC)