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홍차넷
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티타임
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- 다양한 주제에 대해 자유롭게 글을 작성하는 게시판입니다.
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전체기타일상/생각꿀팁/강좌방송/연예도서/문학영화의료/건강문화/예술경제생활체육과학/기술철학/종교정치요리/음식역사여행육아/가정IT/컴퓨터음악창작게임스포츠사회오프모임
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| Date |
Warning: Use of undefined constant reg_date - assumed 'reg_date' (this will throw an Error in a future version of PHP) in /home/redtea/html/a_skin/bbsd/view.php on line 53 17/07/11 14:40:06 Warning: Use of undefined constant mod_date - assumed 'mod_date' (this will throw an Error in a future version of PHP) in /home/redtea/html/a_skin/bbsd/view.php on line 56 |
| Name | 유리소년 |
| Subject | 뫼비우스의 띠에 대한 오해 바로잡기 |
 뫼비우스의 띠는 기다란 직사각형의 한쪽 끝을 반바퀴 꼬아 다른쪽 끝에 붙여서 만들어지는 도형입니다. 동시에 수학적인 도형 중 터무니없는 신비주의에 기반한 잘못된 수사의 도구로 가장 많이 희생되는 도형이지요. 이 글에선 글쓴이의 월도타임을 이용해 뫼비우스의 띠에 대한 잘못된 인식들에 대해 간단히 정리해보도록 하겠습니다. 1. 뫼비우스의 띠는 [무한함]을 상징해요! - 가장 흔히 퍼져있는 오해 중 하나입니다. 하지만 뫼비우스의 띠는 무한과는 별 상관이 없습니다. 뫼비우스의 띠를 만들기 위해 문방구에서 색종이를 살때 색종이가 무한히 들어가진 않잖아요? 2. 뫼비우스의 띠는 [무한한 순환]을 상징해요! - 반만 맞는 이야기입니다. 아무 지점이나 잡고 계속 돌아도 돌아도 원위치니까 무한한 순환이 아니냐고요? 그렇다면, 무한한 순환을 가장 잘 상징할 수 있는 도형은 [원]이겠지요. 우리가 유치원 때부터 배우는 그 동그라미 말이에요. 무한한 순환을 상징하는 물건으로 원 대신에 뫼비우스의 띠를 이야기하는 것이 당신을 딱히 더 유식하게 만들어주지는 않습니다. 마치 어릴 때 산수 시간에 선생님이 "숫자를 하나 말해보세요!" 라고 했을 때 다른 친구들은 3이요! 6이요! 하는데 혼자 2πi요! 하는 것과 같아요. 문과생들에게는 거부감을, 수학도들에게는 비웃음을 살지는 모르겠네요. 3. 뫼비우스의 띠는 [2차원의 특성과 3차원의 특성을 동시에 가진] 오묘한 물건이에요! - 2차원이면 2차원인 거고 3차원이면 3차원인 거지 2차원의 특성과 3차원인 특성을 동시에 갖는 건 세상에 없어요. 먼저 논의 이전에 [도형 자체의 차원]과 [도형을 담고 있는 공간의 차원]을 구별지어서 이야기해야겠어요. 수학자들이 도형 그 자체의 차원을 수학적인 의미로 정할 때는, 그 도형의 [일부분]들이 선, 면, 3차원 공간 등등 가장 잘 알려진 n차원 공간들 중 어떤 것과 가장 비슷한지를 생각해본 뒤 그 도형의 차원을 정하게 돼요. (R^n과 locally homeomorphic하면 n-manifold라고 하지요.) 원의 경우 원의 일부분을 가위로 뚝 잘라서 관찰하면 선을 구부러트린 모양이에요. 그래서 원은 1차원 도형. 속이 빈 공의 경우 그 일부분을 가위로 잘라서 관찰하면 역시 평면을 구부러트린 모양이에요. 그러니까 속이 빈 공은 2차원 도형. 속이 꽉찬 공의 경우 일부분을 가위로 자르면 고체 덩어리이지요. 그러므로 속이 꽉찬 공은 3차원 도형. 뫼비우스의 띠는 평면 직사각형을 꼬아붙여 만든 거니까 일부분을 가위로 잘라도 평면 사각형. 그러니까 뫼비우스의 띠는 [2차원 도형]입니다. 3차원의 특성 관련 이야기가 나오는 이유는 [우리가 뫼비우스의 띠를 만든 공간의 차원이 3차원]이기 때문이겠지요. 하지만 그건 우연히도 문방구에서 종이를 사서 뫼비우스의 띠를 만든 우리가 사는 공간이 3차원이기 때문이지, 뫼비우스의 띠가 3차원의 특성을 가져서가 아니에요. 도형을 담고 있는 공간의 차원은 그 도형을 [어느 공간에 담느냐]에 따라 마음대로 변화하기 때문이지요. 예를 들어서, 원은 2차원 평면에 사는 애니 속 캐릭터들이 2차원 공간 내에서 원을 그릴 수도 있는 것이고, 3차원에 사는 홍차넷 유저들이 3차원 공간 내에서 고무줄로 원을 만들 수도 있는 것이지요. 하지만 그것이 원이 1차원 도형이라는 사실을 바꾸지는 않아요. 남성은 현재 월도짓을 하며 이 글을 쓰는 글쓴이의 본질적인 특성이지만, 글쓴이가 피자집을 갈 때는 피자집의 특성을 부여해 피자집 남성, 직장에 갈 때는 직장의 특성을 부여해 직장 남성 이라고 부르는 건 이상하잖아요? 그 도형이 [현재 어느 공간에 담겨있는지]는 그 도형의 본질적인 특성을 말해주지 않습니다. 4. 그렇군요. 하지만 [뫼비우스의 띠를 담을 수 있는 공간의 최소 차원은 3차원]이잖아요! - 그렇지 않아요. 뫼비우스의 띠를 [2차원 평평한 평면] 내에 담을 수 없는 것은 사실이지만, 2차원 도형 중에서는 우리가 알고 있는 평평한 평면만 있는 게 아니니까요. (위의 예제에서 속이 빈 공이 2차원 도형이라는 예를 보셨겠지요?) 뫼비우스의 띠를 담을 수 있는 2차원 도형 중 가장 쉽게 생각할 수 있는 예가 있어요. 바로 [뫼비우스의 띠 자기 자신]이지요. 모든 집합은 그 자신의 부분집합이니까요. 5. 그렇군요. 하지만 평평한 선, 평평한 면, 평평한 3차원 공간 등등을 나열했을 때 뫼비우스의 띠를 집어넣을 수 있는 가장 작은 공간은 저 중 세번째 것 아닌가요? - 맞아요. 수학적으로는 저것들을 각각 R^1 (직선), R^2 (평면), R^3 (3차원 공간), ...이라고 부르지요. 뫼비우스의 띠는 2차원 도형이면서 평면 R^2에는 매끄럽게 껴묻기(smooth embedding)이 안 되지만, R^3에는 가능하지요. 하지만 그런 특성도 역시 뫼비우스의 띠만의 특성은 아닙니다. 저런 2차원 도형은 아주아주 많아요. 뫼비우스의 띠 말고도 저런 특성을 갖는 2차원 도형들 중 하나는 이미 본문에 등장했는데요, 한번 맞춰보세요. (정답 : 속이 빈 공) 6. 그럼 뫼비우스의 띠는 뭐죠? 뫼비우스의 띠가 다른 흔한 도형과 비교해 가장 차별화되는 특성은 [2차원 곡면이면서 안팎의 구분이 불가능]하다는 점이에요. 이것은 순환이라든지, 무한하다든지와는 그다지 상관이 없습니다. 예전 창세기전 3이라는 게임에서 뫼비우스의 우주라는 괴이쩍은 묘사가 나왔는데, 그거 쓴 시나리오 작가를 누가 한대 때려줘야 한다니까요. Warning: Use of undefined constant no - assumed 'no' (this will throw an Error in a future version of PHP) in /home/redtea/html/a_skin/bbsd/view.php on line 133 Warning: Use of undefined constant no - assumed 'no' (this will throw an Error in a future version of PHP) in /home/redtea/html/a_skin/bbsd/view.php on line 146 Warning: Use of undefined constant vote - assumed 'vote' (this will throw an Error in a future version of PHP) in /home/redtea/html/a_skin/bbsd/view.php on line 161 11
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Warning: Use of undefined constant no - assumed 'no' (this will throw an Error in a future version of PHP) in /home/redtea/html/pb/view.php on line 628
Warning: Use of undefined constant HTTP_HOST - assumed 'HTTP_HOST' (this will throw an Error in a future version of PHP) in /home/redtea/html/pb/view.php on line 640
Warning: Use of undefined constant name - assumed 'name' (this will throw an Error in a future version of PHP) in /home/redtea/html/pb/view.php on line 644
Warning: Use of undefined constant no - assumed 'no' (this will throw an Error in a future version of PHP) in /home/redtea/html/pb/view.php on line 646
Warning: Use of undefined constant no - assumed 'no' (this will throw an Error in a future version of PHP) in /home/redtea/html/pb/view.php on line 646
Warning: Use of undefined constant HTTP_HOST - assumed 'HTTP_HOST' (this will throw an Error in a future version of PHP) in /home/redtea/html/pb/cmt_replace.php on line 3
Warning: Use of undefined constant ismember - assumed 'ismember' (this will throw an Error in a future version of PHP) in /home/redtea/html/a_skin/bbsd/view_comment.php on line 6
Warning: Use of undefined constant file_name - assumed 'file_name' (this will throw an Error in a future version of PHP) in /home/redtea/html/a_skin/bbsd/view_comment.php on line 28
Warning: Use of undefined constant no - assumed 'no' (this will throw an Error in a future version of PHP) in /home/redtea/html/a_skin/bbsd/view_comment.php on line 65
85498 class="iamcmt" style="display:none;"> Warning: Use of undefined constant no - assumed 'no' (this will throw an Error in a future version of PHP) in /home/redtea/html/a_skin/bbsd/view_comment.php on line 66
85498 class='vcm '>
Warning: Use of undefined constant x - assumed 'x' (this will throw an Error in a future version of PHP) in /home/redtea/html/a_skin/bbsd/view_comment.php on line 70
Warning: Use of undefined constant x - assumed 'x' (this will throw an Error in a future version of PHP) in /home/redtea/html/a_skin/bbsd/view_comment.php on line 71
Warning: Use of undefined constant no - assumed 'no' (this will throw an Error in a future version of PHP) in /home/redtea/html/a_skin/bbsd/view_comment.php on line 76
Warning: Use of undefined constant ismember - assumed 'ismember' (this will throw an Error in a future version of PHP) in /home/redtea/html/a_skin/bbsd/view_comment.php on line 117
Warning: Use of undefined constant ismember - assumed 'ismember' (this will throw an Error in a future version of PHP) in /home/redtea/html/a_skin/bbsd/view_comment.php on line 118
Warning: Use of undefined constant ismember - assumed 'ismember' (this will throw an Error in a future version of PHP) in /home/redtea/html/a_skin/bbsd/view_comment.php on line 119
Warning: Use of undefined constant ismember - assumed 'ismember' (this will throw an Error in a future version of PHP) in /home/redtea/html/a_skin/bbsd/view_comment.php on line 120
Warning: Use of undefined constant ismember - assumed 'ismember' (this will throw an Error in a future version of PHP) in /home/redtea/html/a_skin/bbsd/view_comment.php on line 121
85498'> 무적의청솔모
Warning: Use of undefined constant mod_date - assumed 'mod_date' (this will throw an Error in a future version of PHP) in /home/redtea/html/a_skin/bbsd/view_comment.php on line 133
Warning: Use of undefined constant no - assumed 'no' (this will throw an Error in a future version of PHP) in /home/redtea/html/a_skin/bbsd/view_comment.php on line 140
Warning: Use of undefined constant no - assumed 'no' (this will throw an Error in a future version of PHP) in /home/redtea/html/a_skin/bbsd/view_comment.php on line 140
Warning: Use of undefined constant reg_date - assumed 'reg_date' (this will throw an Error in a future version of PHP) in /home/redtea/html/a_skin/bbsd/view_comment.php on line 142
17/07/11 14:44
85498>삭제 주소복사
Warning: Use of undefined constant level - assumed 'level' (this will throw an Error in a future version of PHP) in /home/redtea/html/a_skin/bbsd/view_comment.php on line 150
위상수학의 추억이... ㄷㄷ
Warning: Use of undefined constant level - assumed 'level' (this will throw an Error in a future version of PHP) in /home/redtea/html/a_skin/bbsd/view_comment.php on line 158
Warning: Use of undefined constant password - assumed 'password' (this will throw an Error in a future version of PHP) in /home/redtea/html/a_skin/bbsd/view_comment.php on line 166
Warning: Use of undefined constant vote - assumed 'vote' (this will throw an Error in a future version of PHP) in /home/redtea/html/a_skin/bbsd/view_comment.php on line 171
Warning: Use of undefined constant vote - assumed 'vote' (this will throw an Error in a future version of PHP) in /home/redtea/html/a_skin/bbsd/view_comment.php on line 172
Warning: Use of undefined constant vote - assumed 'vote' (this will throw an Error in a future version of PHP) in /home/redtea/html/a_skin/bbsd/view_comment.php on line 172
Warning: Use of undefined constant no - assumed 'no' (this will throw an Error in a future version of PHP) in /home/redtea/html/a_skin/bbsd/view_comment.php on line 173
Warning: Use of undefined constant no - assumed 'no' (this will throw an Error in a future version of PHP) in /home/redtea/html/a_skin/bbsd/view_comment.php on line 173
Warning: Use of undefined constant vote - assumed 'vote' (this will throw an Error in a future version of PHP) in /home/redtea/html/a_skin/bbsd/view_comment.php on line 173
Warning: Use of undefined constant level - assumed 'level' (this will throw an Error in a future version of PHP) in /home/redtea/html/a_skin/bbsd/view_comment.php on line 158
Warning: Use of undefined constant password - assumed 'password' (this will throw an Error in a future version of PHP) in /home/redtea/html/a_skin/bbsd/view_comment.php on line 166
Warning: Use of undefined constant vote - assumed 'vote' (this will throw an Error in a future version of PHP) in /home/redtea/html/a_skin/bbsd/view_comment.php on line 171
Warning: Use of undefined constant vote - assumed 'vote' (this will throw an Error in a future version of PHP) in /home/redtea/html/a_skin/bbsd/view_comment.php on line 172
Warning: Use of undefined constant vote - assumed 'vote' (this will throw an Error in a future version of PHP) in /home/redtea/html/a_skin/bbsd/view_comment.php on line 172
Warning: Use of undefined constant no - assumed 'no' (this will throw an Error in a future version of PHP) in /home/redtea/html/a_skin/bbsd/view_comment.php on line 173
Warning: Use of undefined constant no - assumed 'no' (this will throw an Error in a future version of PHP) in /home/redtea/html/a_skin/bbsd/view_comment.php on line 173
Warning: Use of undefined constant vote - assumed 'vote' (this will throw an Error in a future version of PHP) in /home/redtea/html/a_skin/bbsd/view_comment.php on line 173
85498>
85498>
