Over the week, I have found a few more things of note to bring to the metaphorical discussion table. Firstly, there is a most welcome patch to the squabble over the proof of necessity of convexity in maximizing area, and secondly, a proposed sketch of a definition of "inside", used when defining area. However, it is still not clear how the definition can be used explicitly in the proof apart from "obvious"-style handwaving at the moment.
Firstly, the patch:
Consider the following 2 propositions:
1) A larger fixed perimeter will yield a larger maximal area.
2) A straight line is the shortest path between 2 fixed points.
For (2), there is not much to say, and is more or less a take-it-or-leave-it axiomatic definition, due in part to the triangle inequality.
For (1), consider 2 fixed perimeters P and P', with P'>P. Consider the shape with the maximal area for P, S. Define S' similarly. Clearly Area(S')>= Area (similar shape to S with perimeter P')* > Area (S), since Area (S)>0 (consider a square, for example). In fact, Area (similar shape to S with perimeter P') = Area (S) * (P'/P) ^2.
Consider a concave (used here possibly loosely to mean non-convex) shape S. Assume it has maximal area.
Then consider of it, two points on its outline A and B such that there exists a point P on line segment AB such that P is not in the shape.
Next, consider the shape S', which is S with arc AB replaced with line segment AB.
Now, consider the shape S'', which is the union of the set of points contained in S and S'.
The area of S'' is clearly greater than that of S, since there is at least one point contained in S'' not contained in S (P), and whatever is contained in S is by definition contained in S''.
3) The perimeter of S'' is also not more than that of S.
<Proof of (3):
Assume otherwise. If so, then there exists an arc CD in between such that its length is less than line CD. This contradicts (2). We are done.>
Hence, S'' has greater area but less than or equal perimeter to S, and hence a shape S''' similar to S'' but with perimeter equal to S with have greater area than S, contradicting the original assumption that S had maximal area for its perimeter.
Finally, on to part 2 of this post, the definition of "inside":
All points at infinite are outside. Consider any line. Each time it intersects the perimeter of the shape indicates a border between "inside" and "outside". The area of a shape is the sum of areas of all "inside" points.
* Similar as in similar figures, used in the same context as congrurent
Thursday, September 27, 2012
Friday, September 21, 2012
Largest area given a fixed perimeter
The question is simple: given a fixed perimeter of say P, find the shape that gives the largest possible area. The answer is somewhat equally simple: a circle gives the largest area for a fixed perimeter. The proof, is not so clear. The definition is a trickier problem, which I shall not care to resolve (I define the center of the circle to be the outside of the perimeter...).
Browsing the internet yields more than just one proof and references to proofs that exist. One of which I trust to be rather unambiguously true is referenced to as the proof by calculus of variations, which I suppose works, since calculus is such an overpowered sledgehammer. However, yet another interesting proof presents itself, of which the outline is presented below:
Define X to be a shape that maximizes the area for a given P.
1. X is convex.
Proof: If X is not convex, then let C be a point outside of X, and A and B be points on the perimeter of X such that C lies on AB. Moreover, let there be no points on the outline of X, say Q, such that Q lies on line segment AB. In that case, if we reflect the outline AB about the line AB, the area increases. This contradicts the maximality of area X. Q.E.D..
Attack: What if we cannot pick A and B such that there are no points of the outline of X, Q, such that Q lies between A and B? (a.k.a. fractals)
2. Any polygon with a fixed number of sides N has maximum area when it is regular. (more or less indisputable).
Corollary: A regular polygon with i sides has greater area than one with j sides given a fixed perimeter iff i>j. (Think of degenerate polygons)
3. Therefore, as the number of sides N goes to infinity, the area increases, and we get a circle, and therefore a circle maximizes area for a fixed perimeter P.
Attack: Where did that come from? Can we even apply rule number 2 to "polygons with infinite sides" (a.k.a. shapes with curves)?
Is there a reasonable patch to the leap from part 2 to part 3? I would agree that it is intuitively very probably true that the argument *sort of* works, and just need formalisation, but with infinities I would not be very sure, ever since the last time I saw 1=1+0+0+...=0+1+0+...=0+0+1+...= ... =0+0+0+...=0. Therefore 1=0. Instead I would probably work towards a more geometry-based idea, described handwaving-ly all the way through, since I have not worked out a full "proof" yet:
Lemma 1 remains, despite the attack. Fractals can be dealt with separately, ... some other time... some other person.
2.The outline of X is smooth. (i.e. no line segments anywhere) random rantings.
Imagine the outline as a wire. Now divide the wire into 2 parts of equal length, and call the parts, with direction, AB and BA respectively (i.e. starting with A and going say clockwise to B and starting with B and going clockwise to A respectively). Define the area enclosed by arc AB and line segment BA be called ABA and BAB be defined similarly. if ABA > BAB, then copy ABA over with symmetry on like AB. X clearly does not have this property since otherwise the operation would produce a shape with the same perimeter and larger area. Hence for X, ABA=BAB for every choice of A and B. Now notice that the reflection does not change the area for X, even though it might change the shape. This does not bring us to an immediate conclusion, since X might not be a unique shape. Also, reflecting about AB must result in another convex shape. This fixes X as a circle... somehow? Good night.
Browsing the internet yields more than just one proof and references to proofs that exist. One of which I trust to be rather unambiguously true is referenced to as the proof by calculus of variations, which I suppose works, since calculus is such an overpowered sledgehammer. However, yet another interesting proof presents itself, of which the outline is presented below:
Define X to be a shape that maximizes the area for a given P.
1. X is convex.
Proof: If X is not convex, then let C be a point outside of X, and A and B be points on the perimeter of X such that C lies on AB. Moreover, let there be no points on the outline of X, say Q, such that Q lies on line segment AB. In that case, if we reflect the outline AB about the line AB, the area increases. This contradicts the maximality of area X. Q.E.D..
Attack: What if we cannot pick A and B such that there are no points of the outline of X, Q, such that Q lies between A and B? (a.k.a. fractals)
2. Any polygon with a fixed number of sides N has maximum area when it is regular. (more or less indisputable).
Corollary: A regular polygon with i sides has greater area than one with j sides given a fixed perimeter iff i>j. (Think of degenerate polygons)
3. Therefore, as the number of sides N goes to infinity, the area increases, and we get a circle, and therefore a circle maximizes area for a fixed perimeter P.
Attack: Where did that come from? Can we even apply rule number 2 to "polygons with infinite sides" (a.k.a. shapes with curves)?
Is there a reasonable patch to the leap from part 2 to part 3? I would agree that it is intuitively very probably true that the argument *sort of* works, and just need formalisation, but with infinities I would not be very sure, ever since the last time I saw 1=1+0+0+...=0+1+0+...=0+0+1+...= ... =0+0+0+...=0. Therefore 1=0. Instead I would probably work towards a more geometry-based idea, described handwaving-ly all the way through, since I have not worked out a full "proof" yet:
Lemma 1 remains, despite the attack. Fractals can be dealt with separately, ... some other time... some other person.
2.
Imagine the outline as a wire. Now divide the wire into 2 parts of equal length, and call the parts, with direction, AB and BA respectively (i.e. starting with A and going say clockwise to B and starting with B and going clockwise to A respectively). Define the area enclosed by arc AB and line segment BA be called ABA and BAB be defined similarly. if ABA > BAB, then copy ABA over with symmetry on like AB. X clearly does not have this property since otherwise the operation would produce a shape with the same perimeter and larger area. Hence for X, ABA=BAB for every choice of A and B. Now notice that the reflection does not change the area for X, even though it might change the shape. This does not bring us to an immediate conclusion, since X might not be a unique shape. Also, reflecting about AB must result in another convex shape. This fixes X as a circle... somehow? Good night.
Friday, August 31, 2012
Constrained Writing
I've been wanting to post this for quite some time. In fact, such some time that I can't exactly remember whether I actually ever got round to writing a post about constrained writing, but knowing me and my laziness, as well as the latest spate of blog-neglect, I shall assume that I never had.
First: before talking about what it was I think I wanted to talk about, I shall acknowledge the main source of my inspiration for this post: namely the Pilish work Cadaeic Cadenza (starting from the poem "Near A Raven"). It is an epic work of constrained writing, mainly that the number of letters of each word corresponds to digits in pi, i.e. Pilish. As if such an undertaking was not enough, the author (writer? composer?) saw fit to add other constraints into his writing through different parts of his story. Ridiculously, the story is coherent and (somewhat) elegantly written, if a bit awkward due to abuse of synonyms at numerous points. As an illustration, "Books inhabited each table, shelf, and nook.". Inhabited? Interesting use of word, of course with this particular instance of "nook". Also, "A madrigal; tell a marcher," seems a little arcane to me.
So, to the point of this post, which at some point became to evaluate the technical niceness of a piece of constrained writing. This is clearly subjective, but nevertheless, some indicators can be objectively evaluated (somewhat like standard of living indicators pointing to the quality of life).
Firstly, of course, there is the freedom that the constraints allow. For example, a constraint that you may not use the word gneiss is hardly a constraint at all under most normal circumstances, since English grammar does not commonly necessitate the use of such a word under most contexts. Writing without using any forms of the word "be", on the other hand, might be a challenge, since there ARE many situations where such a word IS used in the syntax of English.
Then there is, of course, coherence, where the text has to make sense. "For a dime, I would do proper Maths." is, for example, much more coherent than "Fly a poke: a flail or uneven table", despite that the two share the same constraint of Pilish. This is naturally highly subjective. I AM going to be contradicted... right?
Lastly, there is length. A good length makes any piece of constrained writing more impressive, since the whole idea is sort of a challenge upon yourself, and any lengthy piece could be compared to a marathon instead of a 100-metre race, which would be more than 422 times more impressive if traversed at the same velocity.
And once again, I conclude a post hastily, due to sudden lack of inspiration in the midst of composing (I recall there being a more... uncommon *inimitable is a synonym!* word for it). I would nearly apologise for this, but then that necessitates far more apologies to come, which would get old.
First: before talking about what it was I think I wanted to talk about, I shall acknowledge the main source of my inspiration for this post: namely the Pilish work Cadaeic Cadenza (starting from the poem "Near A Raven"). It is an epic work of constrained writing, mainly that the number of letters of each word corresponds to digits in pi, i.e. Pilish. As if such an undertaking was not enough, the author (writer? composer?) saw fit to add other constraints into his writing through different parts of his story. Ridiculously, the story is coherent and (somewhat) elegantly written, if a bit awkward due to abuse of synonyms at numerous points. As an illustration, "Books inhabited each table, shelf, and nook.". Inhabited? Interesting use of word, of course with this particular instance of "nook". Also, "A madrigal; tell a marcher," seems a little arcane to me.
So, to the point of this post, which at some point became to evaluate the technical niceness of a piece of constrained writing. This is clearly subjective, but nevertheless, some indicators can be objectively evaluated (somewhat like standard of living indicators pointing to the quality of life).
Firstly, of course, there is the freedom that the constraints allow. For example, a constraint that you may not use the word gneiss is hardly a constraint at all under most normal circumstances, since English grammar does not commonly necessitate the use of such a word under most contexts. Writing without using any forms of the word "be", on the other hand, might be a challenge, since there ARE many situations where such a word IS used in the syntax of English.
Then there is, of course, coherence, where the text has to make sense. "For a dime, I would do proper Maths." is, for example, much more coherent than "Fly a poke: a flail or uneven table", despite that the two share the same constraint of Pilish. This is naturally highly subjective. I AM going to be contradicted... right?
Lastly, there is length. A good length makes any piece of constrained writing more impressive, since the whole idea is sort of a challenge upon yourself, and any lengthy piece could be compared to a marathon instead of a 100-metre race, which would be more than 422 times more impressive if traversed at the same velocity.
And once again, I conclude a post hastily, due to sudden lack of inspiration in the midst of composing (I recall there being a more... uncommon *inimitable is a synonym!* word for it). I would nearly apologise for this, but then that necessitates far more apologies to come, which would get old.
Monday, August 13, 2012
Anagram a margana
Well, I can't let myself be beaten here. (Despite failing to satisfactorily solve even half of those anagrams ph42 gave...) Here's a fresh sampling of anagram hell.
strobe
sunboule
tacuics (Inaccurate tactics lead to death!)
gneiss (I could've given the other word instead, but nobody knows gneiss...)
cretins
liminiac
shidap
morphias
glintei
All popped into my head while trying to solve those 14 below. So yeah. A response.
strobe
sunboule
tacuics (Inaccurate tactics lead to death!)
gneiss (I could've given the other word instead, but nobody knows gneiss...)
cretins
liminiac
shidap
morphias
glintei
All popped into my head while trying to solve those 14 below. So yeah. A response.
Saturday, August 11, 2012
Anagram List
May be edited to expand it. But meanwhile...
1. Naruto
2. Sungei (Malay for swamp I believe)
3. SacLose
4. ProBash (Warning: obscur-ish word)
5. NaEight
6. EpicSac
7. Tactics (Ah well...)
8. ProBing (Eh already a word. Too self-evident, apparently)
9. Chariot
10. Minister
11. Positional (Eh whaaaaat?)
12. Rating (Again, may be somewhat obscure)
13. Tortoise
14. Silver (3+ solutions)
1. Naruto
2. Sungei (Malay for swamp I believe)
3. SacLose
4. ProBash (Warning: obscur-ish word)
5. NaEight
6. EpicSac
7. Tactics (Ah well...)
8. ProBing (Eh already a word. Too self-evident, apparently)
9. Chariot
10. Minister
11. Positional (Eh whaaaaat?)
12. Rating (Again, may be somewhat obscure)
13. Tortoise
14. Silver (3+ solutions)
Saturday, June 30, 2012
Setting a chess puzzle
I was delighted this week, when I thought I managed to construct a delectable Chinese chess puzzle, pretty much by accident and chance. It contained a somewhat less commonly used theme as well as an accidentally set sacrifice. Happily, I chugged it into the engine for sidelines, and maddeningly, about 3 strange defences materialised out of thin air, defences I had never considered. In the end, I hereby resign to my sad puzzle-setting fate and present you the somewhat correct (but not a mate) puzzle.
Uhh... how do I write the plaintext for it again? Meh.
###AK#PR#
####A##H#
EP#######
##P#P##H#
P####P##H
#########
P#####PPR
####E####
######RP#
R####K###
Red to move and win, of course.
More on how my puzzle failed in a later blogpost, if I remember and can be bothered to make the effort. It's late and I'm braindead too. But in the meantime, enjoy the puzzle.
Just in case I remembered my own notation wrongly, the one presented is as follows:
P -- the most important, powerful, and prevalent piece (Darwin's theory!) on the board.
E -- elephant, minister, whatever. The mascot of Chinese Chess (Xiang Qi)
A -- The advisor, or "shi".
R -- Rook, or "ju", but probably not "che"
K -- The lousiest piece on the board -- never have I lost a game not because of it
H -- Knight (or was it N?). Either way, the "ma".
C -- Not featured in this puzzle, the sneaky cannon, or "pao".
Uhh... how do I write the plaintext for it again? Meh.
###AK#PR#
####A##H#
EP#######
##P#P##H#
P####P##H
#########
P#####PPR
####E####
######RP#
R####K###
Red to move and win, of course.
More on how my puzzle failed in a later blogpost, if I remember and can be bothered to make the effort. It's late and I'm braindead too. But in the meantime, enjoy the puzzle.
Just in case I remembered my own notation wrongly, the one presented is as follows:
P -- the most important, powerful, and prevalent piece (Darwin's theory!) on the board.
E -- elephant, minister, whatever. The mascot of Chinese Chess (Xiang Qi)
A -- The advisor, or "shi".
R -- Rook, or "ju", but probably not "che"
K -- The lousiest piece on the board -- never have I lost a game not because of it
H -- Knight (or was it N?). Either way, the "ma".
C -- Not featured in this puzzle, the sneaky cannon, or "pao".
Saturday, June 16, 2012
A badly played Chinese Chess match
Clearly I haven't been playing Chinese Chess seriously in quite some time; I'm blundering pieces left, right, and center, for the most trivial of reasons. But the thing that takes the cake (it's a lie group!) is that I've not been sacrificing for initiative as much as I probably should. That will take some time to reset. And being cooped up in a place without access to the Internet (well, not much of it) for five days every week (all right, maybe six), I don't think I'll have time for quality chess any time soon. However, I did manage to find an average-ish player on a particular chess website today, and lo and behold! -- here is the chess match, with a slight bit of commentary on why I moved certain moves and my failings.
(As a side excuse, it was a 5 0 game.)
Black: Me Red : ???
1. C2.5 c8.5 2. H2+3 h8+7 3. R1.2 r9+1
4. C8.7 h2+1 5. H8+9 r1.2
At this point I was waiting for the response R9.8 to play C2+4. No I don't remember the theory, but that kind of moves look theory-ish. Also, we probably had deviated from main theory lines by then, so...
6. P9+1 r9.4
7. A4+5 c2.3
At this point I rather liked my setup. Sure I wasn't exactly winning, but it looked like I might have an attack along the right side (in my POV).
8. R2+4 p5+1
The start of my troubles.
9. P5+1
Doh!
9 ... a6+5
?? move. Now I'm in a little bit of trouble. Sigh. Since the formatting looks a bit weird, I guess I'll just take this space to comment on my next few moves. His pao, sitting on the middle, is strong and attacking my king. Even though my cannon is pretty much doing the same thing, I have insufficient coordination, and hence I thought it might be safer to just trade every thing off the board... and my cannon could easily be replaced anyway. Following that, I just threatened to kill his xiang as best as I could, and I would even have happily sacrificed my ma for it, but... he just didn't accept. Aw... so much for being an aggressive maniac.
10. P5+1 c5+5 11. E3+5 c3.5 12. R2+2 r4+7
13. E5-3 r2+4 14. C7.5 c5+5 15. E3+5 r2.5
16. R2.3 h7+5
Yet another occasion that I played a dubious move. Now my xiang dies. And before his. Better was probably R5+3 R3+1 R5=7 R3+2+ A5-6, and now I'm threatening a stupid one move mate, and his shuai must move out, whereupon I can check him and put his king in an extremely dubious place vulnerable to harassment by the side, which is rather sneaky and difficult to defend against. My usual defence against this kind of things is just hoping my opponent doesn't see. Either that or trade off the 'ju's off the board and run away in fear.
17. R3+3 a5-6 18. H3-4 h5+ 19. P3+1 h7+5
Just moving my ma along with the flow. I wanted to go there anyway. Yet the bing move was somewhat necessary to prevent a faster check (which could be fatal since his shuai cannot move).
20. R9.8 h5+7 21. H9+8 h7+9
And I thought I was being obvious. Not obvious enough, apparently :(.
22. H4+3 h9+7
Oops, unstoppable mate out of nowhere in particular. And from this moment on I was just being an irritating troll warlord. But he resigned :(. Aww.
23. K5.4 r5.6 24. A5+4 r4+1
25. K4+1 r6+3 26. K4.5 r4.5 27. K5.6 r6+1
(As a side excuse, it was a 5 0 game.)
Black: Me Red : ???
1. C2.5 c8.5 2. H2+3 h8+7 3. R1.2 r9+1
4. C8.7 h2+1 5. H8+9 r1.2
At this point I was waiting for the response R9.8 to play C2+4. No I don't remember the theory, but that kind of moves look theory-ish. Also, we probably had deviated from main theory lines by then, so...
6. P9+1 r9.4
7. A4+5 c2.3
At this point I rather liked my setup. Sure I wasn't exactly winning, but it looked like I might have an attack along the right side (in my POV).
8. R2+4 p5+1
The start of my troubles.
9. P5+1
Doh!
9 ... a6+5
?? move. Now I'm in a little bit of trouble. Sigh. Since the formatting looks a bit weird, I guess I'll just take this space to comment on my next few moves. His pao, sitting on the middle, is strong and attacking my king. Even though my cannon is pretty much doing the same thing, I have insufficient coordination, and hence I thought it might be safer to just trade every thing off the board... and my cannon could easily be replaced anyway. Following that, I just threatened to kill his xiang as best as I could, and I would even have happily sacrificed my ma for it, but... he just didn't accept. Aw... so much for being an aggressive maniac.
10. P5+1 c5+5 11. E3+5 c3.5 12. R2+2 r4+7
13. E5-3 r2+4 14. C7.5 c5+5 15. E3+5 r2.5
16. R2.3 h7+5
Yet another occasion that I played a dubious move. Now my xiang dies. And before his. Better was probably R5+3 R3+1 R5=7 R3+2+ A5-6, and now I'm threatening a stupid one move mate, and his shuai must move out, whereupon I can check him and put his king in an extremely dubious place vulnerable to harassment by the side, which is rather sneaky and difficult to defend against. My usual defence against this kind of things is just hoping my opponent doesn't see. Either that or trade off the 'ju's off the board and run away in fear.
17. R3+3 a5-6 18. H3-4 h5+ 19. P3+1 h7+5
Just moving my ma along with the flow. I wanted to go there anyway. Yet the bing move was somewhat necessary to prevent a faster check (which could be fatal since his shuai cannot move).
20. R9.8 h5+7 21. H9+8 h7+9
And I thought I was being obvious. Not obvious enough, apparently :(.
22. H4+3 h9+7
Oops, unstoppable mate out of nowhere in particular. And from this moment on I was just being an irritating troll warlord. But he resigned :(. Aww.
23. K5.4 r5.6 24. A5+4 r4+1
25. K4+1 r6+3 26. K4.5 r4.5 27. K5.6 r6+1
0-1
In conclusion, I was still down by a bing at the end. This reflects a deterioration of my skills. *sigh*
Till next time then.
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