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複素数と円の幾何中西敏浩（島根大学大学院自然科学研究科）

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問題

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大円の半径は１，甲円の半径は２分の１

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話の内容（多分）➤ 第1章

複素数の基礎➤ 第2章

サークル・パッキングの数学➤ 第3章

円とフラクタル

➤ 第4章

π は円周率か？

第1章

複素数の基礎複素数

a + bi (a, b は実数，i2 =

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複素数は代数方程式が解をもつように数の世界を拡張することで得られた。
<latexit sha1_base64="Eu2LqA2K1lfDER4+65Wjcc76iZc=">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</latexit>

代数学の基本定理
代数方程式n<latexit sha1_base64="6d6ofUHkTq9HYTRNRk5EoTocsN4=">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</latexit>

a0 x + a1 xn 1(n<latexit sha1_base64="TwIu6thkgXZWXYLBI+AqU/hh3rc=">AAACxXichVLLSuRAFD1mfLaO9uhGcNPYrbgYmpuehTIgiCPo0leroNIkmbInmFRCUt3gNOLWmR+YhSsFEfEzBPEHXPgJojtfG0Fv0gFRGb1FUveeW6fOqVRM37FDRXTeoH1qbGpuaW1LtXd87uxKf+leCL1KYImi5TlesGQaoXBsKYrKVo5Y8gNhuKYjFs31H1F/sSqC0PbkvNrwxaprlKW9ZluGYshLE4aQg8QKyhDIQOfqK885GCiBGJfwoDDK+ctOVOXj8YxK5Mj707KZET6VaZsmHttK6SzlKY7M20RPkiySmPbSByz6k0UtVOCyKcnyFhwWCHkss0WCz9gqaowFnNlxX2ATKeZWeJXgFQaj6/yODracoJLraM8wZlus4vATMDODATqjQ7qmUzqiC3r47161eI/IywbPZp0r/FLX3965+w9ZLs8Kv55Z73pWWMNI7NVm736MRKew6vzq73/Xc99nB2qDtEeX7H+XzumYTyCrN9b+jJjdQYovQH/9ud8mC4W8/i1fmClkx8aTq2hFH/r599AxjDFMYRpF1j3BFW5xp01qrqa0an2p1pBwevAitK0nOAmbpw==</latexit>+ a2 xn 2+ · · · + an 1 x + an = 0

1, a0 = 0, a0 ,..., an は複素数でよい）は必ず，複素数の解をもつ。

第1章

複素数の基礎複素数

a + bi (a, b は実数，i2 =

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Ahlfors (Complex Analysis)

Gauss(Tagebuch)

Whittaker-Watson(Modern Analysis)

Hille(Analytic Function Theory)

Bourbaki(Éléments de Mathématique)高木貞治（解析概論）

辻正次（複素函数論）
楠幸男（解析函数論）
岸正倫・藤本坦孝（複素関数論）
岩波

数学辞典（複素数の項目）

Wikipedia -複素数 （仏・西・韓）Wikipedia - 複素数（日・英・独・露・中）複素数の基礎（複素平面と極形式）
xy-平面の点 (x, y) で複素数 z = x + iy で表すとき，この平面を複素平面 (高校の教科書では「複素数平面」) という。 z = x + iy(= 0) は次の表現をもつ。
<latexit sha1_base64="YhLh6G4S5IpbJbNwC6MPllrGfkc=">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</latexit><latexit sha1_base64="bg4o+Y9uYqORxZ7TJ4r6KnkJbLE=">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</latexit>z = r(cos + i sin )<latexit sha1_base64="1nWJtk0oCQXiWb4CJhJ962xTecU=">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</latexit>

これを z の極形式という。
<latexit sha1_base64="DfdaBPyJozdgDaNShWuWTtX7OGE=">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</latexit><latexit sha1_base64="TFqxT1guSxbMWR+GU/0rscKdzTo=">AAADV3ichVK7bhNBFD27hmDMww40SDQhdhARknXXFDwkpAgQShnbOIlkG8u7ntgr70u7Y4NjpaFCiCoNBRVIFIi/gIYfoPAnIMog0SDgzngLQkCM5Zk7Z+ace+7ctSPPTSTRzDAzx44vnMiezJ06feZsvrB4bjMJR7EjGk7ohfG23U2E5waiIV3pie0oFl3f9sSWPbyrzrfGIk7cMHggJ5Fo+91+4O64TlcyFBYeooUAIVyeexA8S+SwghJ2cRuPcZVPJriS3pKMEVZRonB4iYKng+QH7dMT6pGpWS00NS/WDIcZCa8SA1aW6Gq1FmMq2+/4Ku/aSoHq9Jz2dfYSBe6oY3fyNKB7dF9nKB1ilTqFIpVJj6WjgZUGRaRjIyy8Z4Ee23Iwgp+W68BjsYR/TVioIGKsjSljMUeuPhfY4/TKukrtaWZfW5lqS4KfSuIR3+5pdI/xm8z0OWpxLsWLWNHBkOc+75opGvBeOUl0TvVkHv9j5i3xc3yit3RAH+kdfabv2sHftKZaQ1Uw4dWec0XUyT+7UP/2X5bP69xzyvonQ3mW2MEN7VU1MdKIqsKZ88e7Lw7qt2or08v0mr6w/1c0ow9cQTD+6rypitpL5Lht1p9NOhpsVsrWtXKlWimu3UkbmMVFLPOnZeE61rCODTTgGMvGulE1aubM/JlZyGTnV00j5ZzHoZFZ/AUYS7p0</latexit><latexit sha1_base64="2MCbgoKHMCyo1G9TI0g9vujH4y0=">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</latexit>

0=<latexit sha1_base64="cnfeqizKC1qq0cXkvAgJsTH6+O4=">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</latexit>r = |z| =x2 + y 2 を z の絶対値,

を z の偏角という。
偏角は 2 の整数倍を除いて定まる。
<latexit sha1_base64="gziTw9pRtRT+Hir3Fek9mVQbN5k=">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</latexit><latexit sha1_base64="0i/oWOAWHt6TNPpptGQriLWZuxM=">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</latexit>

複素数の基礎（複素数のたし算）
z = a + bi, w = c + di のとき<latexit sha1_base64="DZM36kyS9BTH9R4nJ0eQ1EBcvV4=">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</latexit><latexit sha1_base64="nxayBLzHu/FDK5RqYpQuSzjs2hE=">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</latexit>z ± w = (a ± c) + (b ± d)i

複号同順複素数の基礎（複素数のかけ算）
z = r(cos + i sin ),のとき，
w = R(cos + i sin )<latexit sha1_base64="Y8E0pRV0BbpoMG6JgOQFwR097hA=">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</latexit>

zw = rR(cos( + ) + i sin( + )).
<latexit sha1_base64="wtEhEkQjzC0X87UVMxRtwuU94Vw=">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</latexit>

つまり，絶対値は「かけ算」に，偏角は「たし算」に。
<latexit sha1_base64="1y1j01dgjILDrf0WPvaWnFrxHq8=">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</latexit>

とくに w = cos + i sin のとき，zw は z を原点を中心に角 だけ反時計回りに回転させたものである。
<latexit sha1_base64="JyC3ChjKQDoQj897zsdexnfBvbM=">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</latexit>

w = c + di = 0 のとき
z
z w̄
=
w
|w|2
<latexit sha1_base64="lmdExoM0R6bxn3Zf+sN3nNiXc+8=">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</latexit>

割り算<latexit sha1_base64="d8biXNi1Li9Hri7bB2Cny9hL2bQ=">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</latexit>w̄ = c<latexit sha1_base64="tj6KhniTqqAY/VmbRIYbBsbKJoU=">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</latexit>

di は w の複素共役

複素数の基礎（オイラーの公式）
ie<latexit sha1_base64="Qs69Ba29V8RACMcdOawZc6Fv8x8=">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</latexit>= cos + i sinとくにi<latexit sha1_base64="3rd2FUlEUCdvn/TkGlV4a2NXjlY=">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</latexit>

e =

1

i

<latexit sha1_base64="QYQegt70H2nYkGLB4Nxnxjd6s/o=">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</latexit>

e +1=0オイラーの公式は暗闇に光る一筋の流星だった。暗黒の洞窟に刻まれた詩の一行だった。
小川洋子「博士の愛した数式」より

複素数の基礎(一次分数変換)
し

）

返

り

像

(折

鏡

平

鏡

（
像

返
り
折

行

移

動

）
し

2つの平行な直線に関する鏡像の合成は平行移動

2つの交わる直線に関する鏡像の合成は回転移動

回転移動
鏡像

鏡像

回転角は２直線の交角の２倍円に関する鏡像（反転）円 |za| = r に関する鏡像<latexit sha1_base64="9Y+KGwBO8Lx4mu5Ob+fgS6NZs0w=">AAACi3ichVE9LARREP6sv3P+Do1Ec3GIxmXuFERIhEiUdzg/QWR3vWNze7ub3b1LztEq0IkoVCQiotPSaXQqhVIpShKNwtzeJoJgNvvmm+/NNzPvPcXSNccleqiQKquqa2oDdcH6hsam5lBL66xj5mxVpFRTN+15RXaErhki5WquLuYtW8hZRRdzSma8tD+XF7ajmcaMW7DEclZeM7S0psouU2aoLbmNMLqwiQ30QWY/ApvjMGUOTmiBdvcfE3cIroQiFCXPwj9BzAcR+JYwQ2dYwipMqMghCwEDLmOdWzj8LSIGgsXcMorM2Yw0b19gC0HW5jhLcIbMbIbXNY4WfdbguFTT8dQqd9H5t1kZRjfd0zm90C1d0BO9/1qr6NUozVJgr5S1wlpp3mmffvtXlWXvYv1T9efMLtIY9GbVeHbLY0qnUMv6/Mbhy/TQVHexh07omec/pge64RMY+Vf1NCmmjrwHiH2/7p9gNh6N9UfjyXhkdMx/igA60Ilevu8BjGISCaS47x4ucYVrqVHql4ak4XKqVOFr2vDFpIkP5U6T5A==</latexit>

P<latexit sha1_base64="YIyT6VKSfKHRXPVHzp3vQNiKWZM=">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</latexit><latexit sha1_base64="qc9MmjWYYovr29wA5gD64qhnQFo=">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</latexit>Pz<latexit sha1_base64="uzF41bAyeAJq1ymsAkac3IlT6Vc=">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</latexit>az = x + iy((x, y, 0) と同一視)

z<latexit sha1_base64="MdiHhoUXTSgdzjcjXZ+44mQxjAs=">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</latexit><latexit sha1_base64="HUkYq/r1/TrTpaqb0ogTi51XVs8=">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</latexit><latexit sha1_base64="SjOmdDj7RVX+azUGNugFLUlZ+kM=">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</latexit>

P (P と赤道に関して対称)<latexit sha1_base64="8g7F8ODSJ0nYDcaG/xP8HV1ZC3s=">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</latexit>z = R(z) =<latexit sha1_base64="MjTcYTU6sUk8p9Lt31n5+fQxGmE=">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</latexit>

r2
z

a

+a

無限遠点
を導入して
R(a) = , R( ) = a と定める。
<latexit sha1_base64="HO/YsOZwBdlosZUZ/mf4qX41h/Y=">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</latexit>

円に関する鏡像（反転）<latexit sha1_base64="Fi8ahSMR24/im9G5F0GTa1IdU+4=">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</latexit><latexit sha1_base64="kH/Ekj7jjdL/nR/tsNpyjTY1mMM=">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</latexit><latexit sha1_base64="WFLJkUWW1AYapCzwZT/dkm2Xa+8=">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</latexit>|z<latexit sha1_base64="Z4sM0AW/2JI9m0zma4xRvehAaG8=">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</latexit>

z

z

a

a||z

a| = r2

複素数の基礎(一次分数変換)
以下，直線も「円」（無限遠点を通る円）とみなす。
<latexit sha1_base64="d4ku2WNwrgheNeX2Baw2ZJEcOtc=">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</latexit>

２つの円に関する反転の合成は<latexit sha1_base64="q/R+kPURT+OjYQRR5c41A3uUumk=">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</latexit>

az + bS(z) =
cz + d(adbc = 0)<latexit sha1_base64="G7rNebdrvb+Z5QcOtfai3lmbFNo=">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</latexit><latexit sha1_base64="gOLCJOVdMz9nj8d288FI58XqwN4=">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</latexit>とかける。これを一次分数変換またはメビウス変換という。
<latexit sha1_base64="LFbuppfTsaeDSiH3L3dFxV5hBwY=">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</latexit>S( d/c) =S( ) =, S() = a/c(c = 0)(c = 0)<latexit sha1_base64="GjNFStzKJw3ZHm/gfWN0O4FaNrI=">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</latexit>と定める。
<latexit sha1_base64="gTqi8WiPWLuOu/AygqnN7TJMbDM=">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</latexit>

a, b, c, d に同時に同じ数をかけても変換 S(z) は変わらない。よって adといった条件を課すことがある。
<latexit sha1_base64="XPfjv72eeLg73bCBobeuaSbLRdE=">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</latexit>

bc = 1複素数の基礎(一次分数変換)
az + bS(z) =,
cz + dを一次分数変換とする。

pz + qT (z) =
rz + s<latexit sha1_base64="WGXdWhcNJEHwvF2AYCT6HD8uxSU=">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</latexit>• 一次分数変換の合成は一次分数変換(ap + br)z + (aq + bs)
S(T (z)) =(cp + dr)z + (cq + ds)• 一次分数変換の逆変換は一次分数変換<latexit sha1_base64="vGxwiTXgFoP0c0Uf3mufyKwaQQU=">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</latexit>

S

1

dz b
(z) =
cz + a• 恒等変換は一次分数変換<latexit sha1_base64="RXxbyym2vI5t770skBNK+4LYfOY=">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</latexit><latexit sha1_base64="6hcvIPFjgJFXPpFY7mJaiBbEIhU=">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</latexit>

1·z+0I(z) = z =0·z+1

一次分数変換全体は「群」をなす。

複素数の基礎(立体射影)<latexit sha1_base64="ASeB+75yz9dgpLWapNcRN4fC+us=">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</latexit>C{} は立体射影を通して球面と同一視する。

x 2 + y 2 + t2 = 1<latexit sha1_base64="3OEp6bYhphltHKY2GlxKFFIe9w0=">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</latexit>と同一視
z=x + iy((x, y, 0) と同一視)<latexit sha1_base64="2UIcjrutzB1Zl2OZpfw4ZLo7/Zs=">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</latexit><latexit sha1_base64="SjOmdDj7RVX+azUGNugFLUlZ+kM=">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</latexit>

このとき，C<latexit sha1_base64="5h27QMqh21vj9xGe/X3QyoR5mVk=">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</latexit>{ } をリーマン球面という。

複素数の基礎(一次分数変換)• 一次分数変換はリーマン球面からリーマン球面への1 対 1 対応を与える（正則同型になる。）• 相似変換（拡大・縮小）は２つの同心円に関する反転の合成• 一次分数変換は円を円に写す。
<latexit sha1_base64="OjpSm1UIbKd+phWrCQp84BA7H2o=">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</latexit>

第2章

サークル・パッキングの数学

和田昌昭氏(大阪大学)作成のOPTiを用いて描画上半平面のサークルパッキング

上半平面のサークルパッキングp以下，有理数 は既約分数で表示し，q > 0 とする。
qp1
有理数 で実軸に接し，半径 2 をもつ上半平面q2q
内の (開）円板を C pq とおく。
<latexit sha1_base64="qxOtBdMZb2AXIHyiusKu6xdnOkc=">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</latexit><latexit sha1_base64="GD0C2PkRf4ZT2aIqtDEo+ZBJDa8=">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</latexit>

C rs<latexit sha1_base64="fG9LtYlX03mbEDrkCfSMeUvQq4w=">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</latexit>

C pq<latexit sha1_base64="R9q4kCFVwujxD+6J/U1jemtEx0A=">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</latexit>pr= のとき，C pq と C rs は交わらない。またqs|ps qr| = 1 のときに限り，2 つの円板は接する。
<latexit sha1_base64="+LoLVbyv2PnwSdVkHR3imALX1TM=">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</latexit>上半平面のサークルパッキング<latexit sha1_base64="Xe0t3xB7/i088yJ8Iu5Ec+xQhoo=">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</latexit><latexit sha1_base64="Egf17Bx/KLRifre6CZl/46ba3Kw=">AAACuHichVE9T9tAGH5woYWUNmlZKrEgoqAujc7J0IoJwcLIRwNIcZTal0t6wj5b9pEWrAys/AGGTq2EEGKhv4GFP9CBn1AxUqlLh76+eGkD4rV897zPvc/dc/d6kS8Tzdj1mPVofOLxk8mpwtPpZ8+LpRcvt5JwL+aiwUM/jHc8NxG+VKKhpfbFThQLN/B8se3trmTr230RJzJU7/V+JFqB21OyK7mriWqXZh0VStURShec5hunG7s8tQdpfeC0Cu1SmVWZiblRYOegjDzWwtIlHHQQgmMPAQQUNGEfLhL6mrBRQ0RcCylxMSFp1gUGKJA2IaRNrtAj9JHqHJoFPtP4iao7hh0Qb4ORNiDs0GmZMqI9OXZp7FHWzFlFeeYlMadycufTH5NuDhX2g52xW3bFztlP9sd4uGuv1OyR3WGfZm+oFVG7ePRq8/eDqoDmoetcda8i86zRxTvjVZL3yDDZLfhQ3z84vt1c3KikC+wbuyH/X9k1u6QbqP4vfrIuNr4ga5z9f5tGwVatatertfVaeWk5b+EkZjGP1/S2b7GEVayhQece4hQX+G4tWh+sniWHpdZYrpnBP2HFfwHrt6Ay</latexit>

13<latexit sha1_base64="KizNe6T4OaKRLBG3B/5eCJpO/QM=">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</latexit>

i

0
1

1+i
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1 1
4 3<latexit sha1_base64="DMDfIKzkujP1WpAbN1qJyM6EWTo=">AAACt3ichVG7TtxAFD0YQsjyWkgTQYNYgahW46UgUKGkoeSRBaT1amXPzi4j7LGxh+VhrZQ6P5CCKpFAQhTkH2jyAyn4hCglSDQpcj3rhpCIa3nm3DP3zJyZ60W+TDRjt31W/8CLwZdDrwrDI6Nj48WJye0kPIy5qPLQD+Ndz02EL5Woaql9sRvFwg08X+x4+++z9Z2OiBMZqg/6JBL1wG0r2ZLc1UQ1ilOOCqVqCqULTs1pxS5P7W662HXqhUaxxMrMxMxTYOeghDzWw+INHDQRguMQAQQUNGEfLhL6arBRQURcHSlxMSFp1gW6KJA2IaRNrtAmtEd1Ds0CxzQeUXXTsF3ibTDSBoQdOi1TRrQnxz6NbcpqOasoz7wk5lRO7nz6Y9LNYI79YJfsjn1nV+wn+208/Guv1OyR3eGEZq+nFVFj/NObrYdnVQHNPde56r+KzLNGC2+NV0neI8Nkt+A9fef0893WyuZcOs++sl/k/wu7ZTd0A9W55+cbYvMMWePsv9v0FGxXyvZiubJRKa2+y1s4hGnMYoHedgmrWMM6qnTuR1zgGt+sZathtay9XqnVl2te41FYB38AWwSf+w==</latexit><latexit sha1_base64="aI5WgrzLfD1ijSO+FNpsBd2ll0Y=">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</latexit>

12<latexit sha1_base64="rWzWtVDTXORscFTG5fIzVfj8Rh0=">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</latexit>

23<latexit sha1_base64="0yVFZQudG8vwkBevKr+gCt8+j4o=">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</latexit>

11<latexit sha1_base64="t/69SYYotUbXidAsYclOHxILzZE=">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</latexit>

32<latexit sha1_base64="mGYK0+MAXh/l3qrSMGrunT7O/X8=">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</latexit>

2
1

<latexit sha1_base64="ZfdLCSsPmXRPbexy7dPlCAuf/E8=">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</latexit>

互いに接する直線と 2 円のすべてに接する円の半径abと は既約分数. c > 0,d > 0, ad − bc = ±1.
cd<latexit sha1_base64="bG9wSI8LN4zkUa1lTHyZLTBZxpM=">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</latexit>abと の中間数cdCb<latexit sha1_base64="KdgaBYZnPRtK++yAAjsxKghC0hw=">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</latexit>dC ac<latexit sha1_base64="lBUmECmUYNPdmXttB1ghvJ4jk+k=">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</latexit>

実軸ac<latexit sha1_base64="86ygDVW/bOfM/zW/d41mC0CsFrw=">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</latexit>

b
と仮に表す
d

<latexit sha1_base64="OldV1z3R0LZ6BAWabMyZrj/g5h8=">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</latexit>

0
1

1
1
=
1
2

3
5
8
13
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1
2

1
2
=
1
3

1
2
3
5

2
3
=
3
5
2
5
=
3
8

フィボナッチ数列<latexit sha1_base64="oT9OB8BRiJFU1mSF6JFRxfDNHjA=">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</latexit><latexit sha1_base64="oQNXb8b90eZkP4ukuw5KDFIiGH4=">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</latexit>

5
8
=
8
13
5
13
=
8
21
<latexit sha1_base64="SKhfHNHwkZdX/68QHpc2cJvh+8Q=">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</latexit><latexit sha1_base64="V2OvNLZkcmGsEkeNkQ4YLtn10eY=">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</latexit>

1+ 5
黄金数
の逆数に収束2

1, 1, 2, 3, 5, 8, 13, 21, · · ·フィボナッチ数列<latexit sha1_base64="RTIrdPQrQSw5vE19ZFzTi9j252I=">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</latexit>

a0 = 0, a1 = 1,
an+1 = an + an 1 (n = 1, 2, 3, · · · )<latexit sha1_base64="wP+RYAsTlvQjByXF6JKPMLHEPJU=">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</latexit><latexit sha1_base64="NsaC9VTkZbwKvusH2zbdktQDCPg=">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</latexit>

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, · · ·
50

25

45

30

40

0=55

0=34

35

20
30
5

5

25

15
25
10

10
15

20

サークルパッキングの応用：有理数と無理数の見分け方
実数<latexit sha1_base64="zONITxdaXs+0sa+yXbHuEJfVON8=">AAACvXichVE9T9tQFD2YEmjaQoClEgsQQAwougkDiAUKC0NVQdIAUkCRbR7BwvEz9kv4iDIwFn4AQwdEpaqq+A8sCKl/oAN7lwrRBQRLh944VhEg4Fp+9/Pce959hmtbviI6a9AaXzRFmlteRl+9ftPaFmvvmPNlyTNF1pS29BYM3Re25YisspQtFlxP6EXDFvPG2lQtP18Wnm9J56PacsVSUS841opl6opDMjaUOxAu+rAIHTZcrLLuo/fsKwhs8mlCckbCQ4V/gWVUUVnvURHGValAk7RHedohJ3/6aSfzLh0p6B+uSeZjcUpQIN0PjWRoxBHKjIx946HLPMhECUUe5ATDbSbk85dDEsQEFZaYiM5UFKwgL5hQlLElrhJcoXN0jc8Ce7kw6rBf6+kH6NsrVdGNfvpJ3+mSftAR/aa/j/aqBD1qXLZYG3WscPNtu28zN8+iiqwVr/c/6knOCisYDbhazN0NIrVbmHV8eXv/MjOW7q8M0Bc6Z/6HdEYnfAOnfGV+nRXpz4jyAyTvr/uhMZdKJIcTqdlUfGIyfIoWdKEXg7zvEUxgGjPI8txj/MIF/mjjmtBszamXag0hphN3RNv4B9jNo6Y=</latexit>

が無理数であるための必要十分条件は

p
q

1
2q 2

<latexit sha1_base64="JTBJ0AXxLLqgMCQJlJNhC+pow9Y=">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</latexit>pをみたす有理数 (既約分数, q > 0) が無限個q存在することである。
<latexit sha1_base64="9ZzuE356HO0nXu44VD/U5upgNGo=">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</latexit>

証明

p
q

1
2q 2

<latexit sha1_base64="JTBJ0AXxLLqgMCQJlJNhC+pow9Y=">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</latexit>

を通る垂直線が円板 Cp/q と交わる。
<latexit sha1_base64="qEQp2Syt0syb90damOX0Pl7J/p4=">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</latexit>

<latexit sha1_base64="f3q7afj1A14qHiUm9T/cz8atMRk=">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</latexit>

が有理数のとき
Cp/q がオレンジの円板のときに限り<latexit sha1_base64="siVEksvnirdSJKmHY2wxGbBmzYg=">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</latexit>

p
1
< 2
q
2q
<latexit sha1_base64="+fe3Y1txw4lTd7QrX1exfXGy2AQ=">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</latexit>p
そのような有理数 は有限個しかない。
q<latexit sha1_base64="FS7YSyzLcYiMbAl9zIBW10a0inw=">AAADC3ichVExb9NQEL64pTQp0AALEktFWsQUncNQ1KkChJBY0pY0lZqosl9ejVvb79V+DYQoO0JihYEJJAbEyIbYKiR+AAz9CYixSEiIgc8vXqAgnuXv3Z3vu/vO5+sozAzzYcmZmDwxdXK6XJk5dfrMbPXsufVM7adCtoSKVLrhe5mMwkS2TGgiuaFT6cV+JNv+7o38e7sv0yxUyV0z0LIbe0ESbofCMwip6iJ1KCFFIbBHEmiowl1O+BHf4p2gbKakpnkaIq+HrIw0ReTRAJYBRuB0aJtSxASyNI2Ae8ARzbMKyk+bzRlu8x3e4ZvsbFVrXGd75o4bbmHUqDhNVT2wjRWK71NcCBRWQoZnk1xqoKmhLtp6kGEgUlhZI6qAm8Ey1k8ogHXPjmLgPwDeR3bPRnPZLjG4MewOuuVMbcfaBQbwNotoAj/XktmuAuoivCl4c7TAn/k1H/FHfsNf+KfV8LdaQ1sjn2GA2x9zpd6afXxh7ft/WTHuseqC9U9GrtlgQdes1nzN2kbyKcSY33/47GhtaXVheJlf8lfof8GHfIAJkv438WpFrj6nChbn/rmm48Z6o+5erTdWGrXl68UKp+kiXaIr+LeLtEy3qUkt9P1EP0oTpUnnifPWeee8H6c6pYJznn47zodfM7KwKA==</latexit>

α=

r
s

が無理数ならば，垂直直線に沿って に近づくとき，いが無理数ならば，垂直線に沿ってに近づくとつまでも Cp/qCの形の円板に入ったり出たりを繰り返す。
いつまでも
p/q の形の円板を通り抜け続ける。
<latexit sha1_base64="hID95Frb17jVAcKlLdW3Az7tqbM=">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</latexit><latexit sha1_base64="PaYEvMs4SSKBkM0M5MWE4xYdcS8=">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</latexit>e<latexit sha1_base64="coMFln5ri5qcIgqRGH9/gWcGEos=">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</latexit>

証明終わり

問題の解答38

27

半径の逆数を示す

18

11

2

6

乙

丙

3

6

<latexit sha1_base64="zorzA/BFULp0AfnocY4x61sRMtI=">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</latexit><latexit sha1_base64="SRt+6N7DsVpBoJ4G0zcweFYqvcA=">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</latexit>

3

丁

戊

己

庚

11

18

27

38

規則性を見つけてください！

問題の解答 互いに接する3円に接する円の半径（デカルトの円定理）<latexit sha1_base64="nOHqdhTrsaDoGMKDabg9ewfCZHM=">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</latexit>

1
1
1
1
=
+
+
+2
r
r1
r2
r3

1
1
1
+
+
r2 r3
r3 r1
r1 r2

1
=
r
<latexit sha1_base64="s19JLmI78DJUTEL4jShVk4+yK2w=">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</latexit>

1
r1

1
r2

1
+2
r3

1
1
1
+
+
r2 r3
r3 r1
r1 r2

男の児が中学(注：旧制中学)で幾何や代数を習うのは何の為めか、
必ずしも実用に供するのが主眼ではなく、頭脳の働きを緻密にし、錬磨するのが目的ではないか。
谷崎潤一郎「痴人の愛」より

解答はデカルトの円定理を使った数学的帰納法でも得られるが，図の一次分数変換による Cn の像の半径を計算してもよい。
<latexit sha1_base64="MYJyWqoof/ihF4RgtBjqe612ckc=">AAADJ3ichVJNaxNRFL0ZNU1ja6OCCC6MphVX4SQuFFfFKoiI5qNJKkkdkpeXOGQyM8xMozUEoRSxratuArpScFH6M7rQXVcRiytxJW6ECG4EvTOZjVbpG+a+8867577zPmqWrjkusBdSDh0+Eh6LjEePTkwem4odP1F0zCVbyIIwddNeqFUdqWuGLLiaq8sFy5bVdk2XpVprzpsvdaTtaKYx7y5bcrFdbRpaQxNVlykzdoMqZJBJGsc6SY4uRdc+d+dh4i228RXvYGSfll+4YWzefQIBVVrlRmf1PnqT+gBNrKh9rOE5NnD710RxCOMOtc7lwtK6Ny4+ooVn2KBpmiOVa0/DyOzCyN9SV7G5Hun2UUIDK5xzHYoaSyAJv8X3g1QAEhS0jBnbYfN1Ni9oidqBeUE6Vcnhr0wpSpPF3CJ1mbMZaf68pB5FWeswcv2xQU1GDzivwr2kRxwfcnbdZ3vMpwisbTOu8Gqe0uKaglocmzwqB6zBY8+L468q2J3Ov826OM1ggC0M+Ui38QU/fQ//qtX1a3h7WOa+NtJKS51aP53/caCqzf3IdaD6r8Lz7FKDrvhevSdg+Yy3CzHSdx73h/mruZnuBbzmxzDAK+xhx3sUne/iTVbmXlKULy719zXtB8V0MnUpmc6mE7PXgiuM0Bk6Txf5bC/TLN2kDBV43W+hU6Gzobiypewq75UPo1QlFGhO0h9N+fQbNdHE+Q==</latexit>

0

1

2

3

4

1

1

1

1

1

( 1 + i)z + (1 + i)
S(z) =(1 + i)z + ( 1 + i)
<latexit sha1_base64="olwTR1r40Z7pEmq0hsM9Gq+Nu7g=">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</latexit>

3

6

11

18

27

38

1

4

9

16

25

36

<latexit sha1_base64="j/cy7v0r2Dfx3ZCZcBemfp54rKc=">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</latexit>

2 を引く解答はデカルトの円定理を使った数学的帰納法でも得られるが，図の一次分数変換による Cn の像の半径を計算してもよい。
<latexit sha1_base64="MYJyWqoof/ihF4RgtBjqe612ckc=">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</latexit>

0

1

2

3

4

1

1

1

1

1

( 1 + i)z + (1 + i)
S(z) =(1 + i)z + ( 1 + i)

S による C pq の像は円で，半径の逆数は<latexit sha1_base64="olwTR1r40Z7pEmq0hsM9Gq+Nu7g=">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</latexit>

2<latexit sha1_base64="yfc5f4bm+hhjBNwNNEpJaYTOg3Y=">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</latexit>2

p +q +127

半径の逆数を示す
74

59

18

54

30

38

51

66

83

146171102123

66

51

38

27

63

11

18

54

63

11

30

59

6

66

35

59

62 23

47

74

39

2

59
14

83

95

6

35

12310295

3966

171146

23 62
50

47

14

50

26

26

42

42

62

62

86

86

114

114

3

38

3

38
15

35 6399143

143
9963 35

15

38

38

114

114

86

86

62

62

42

42

26

50

14

62 23

47

23 62

59
35

39
59

30
74

11
54

95

95

38

51

66 83

102123146171

66

39

63

27

14
35

6

63

18

47

59

2

6

66

26

50

123102171146

83

66

51

38

27

18

11
54

30
74

59

37

50

65

82

101

122 145170

170145

122

101

82

65

50

37

半径の逆数から1を引く
17
53

26

26

94

94

62
73

58

29

17

53

62

10

10

73

29

58
<latexit sha1_base64="jkXdWPOG0zyKbvc6SqBq7c+TjA4=">AAACa3ichVFNLwNBGH66vqo+WnoRHERbcWqm2wNxarg40ipNENldg4n9yu60CY0/4OjiUBcSEfEzXPwBBz9BxInExcG7200EwbuZnWeeeZ93nnlHd03hS8YeYkpHZ1d3T7w30dc/MJhMDQ2v+k7dM3jVcEzHq+maz01h86oU0uQ11+OapZt8Td9fCPbXGtzzhWOvyAOXb1rari12hKFJomrZrFosZrOJrVSG5VkYEz9BIQIZRLHkpK6wgW04MFCHBQ4bkrAJDT596yiAwSVuE03iPEIi3Oc4QoK0dcrilKERu0//XVqtR6xN66CmH6oNOsWk4ZFyAjl2z67ZC7tjN+yRvf9aqxnWCLwc0Ky3tdzdSh6PVN7+VVk0S+x9qv70LLGD2dCrIO9uyAS3MNr6xuHpS2WunGtOsQv2RP7P2QO7pRvYjVfjcpmXWwgeoPC93T/BqpovFPPqspopzUdPEccYJjFN/Z5BCYtYQjXs8wlaOIs9K2llVBlvpyqxSJPGl1ByH3WQivo=</latexit>

38

1

5

65
233

38

34

89

58

5

マルコフ数
1
233
2
610
5
89
13
29
34
89
169
194
233
433
610
..
.
<latexit sha1_base64="bMS0sowiWBgxgqIuh/KNtmBPRMc=">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</latexit>

65

34

233

89

58

<latexit sha1_base64="yMLNoVTBpdSp9UXHR7dHFaPTtPs=">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</latexit>

13
434
74

46

61

22

22
49

46

61

13
434
74

49

25

25

41

41

61

61

85

85

113

113

37

2

37
14

34 62 98142

14298 62

34

2

14

37

37

113

113

85

85

61

61

41

41
25

25
49

74
434

13

61

46

49

22

22

58

58

1

89
233

34

5

65

29
73

10
53

46

5

62

62

94

94

26

26
37

50

65

82

101

122 145

170

13
89

65

38

17

74
434

34

38
58

61

122
170145

101

82

65

50

37

17

10
53

29
73

58

233

<latexit sha1_base64="Amk+AiuUDK60KKRDEGUEyvAxcNY=">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</latexit>

170

145

122
101
82

65

50

37
26

94

17

194
169

53

62
73

10

29
58

38

5

65

34
233

433

89

58

22

13
61

46

433
74

49

25

41
61
85
113

37
<latexit sha1_base64="Amk+AiuUDK60KKRDEGUEyvAxcNY=">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</latexit>

1
2
5
13
29
34
89
169
194
233
433
610
..
.

マルコフ数
正整数の３つ組 (x, y, z) が2

<latexit sha1_base64="aaTtu0Ys58Rdt5D8b3VgnbEeZ90=">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</latexit>

x +y

2

22
2
2
x
+
y
+
z
+ z = 3xyz= 3xyz

をみたすとき，マルコフの３つ組 (Markov triple) という。という。
<latexit sha1_base64="4oPMVjCZop6+s714WjAstk5BzS0=">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</latexit><latexit sha1_base64="erjlHSAsmuFvpBNzn74RSHOeL64=">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</latexit>(1, 1, 1) から次の操作を繰り返してすべてのマルコフの３つ組が得られる。
<latexit sha1_base64="itShPDsIfX6G1lSr/r0oLu+iAvc=">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</latexit>(a, c, 3acb)(b, c, 3bc

a)(a, b, c)
(a

b

c)

<latexit sha1_base64="P7pdZdV6zjY2W26gvu1ko6zhBIg=">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</latexit><latexit sha1_base64="Td+S2sA7ELPBlaPPUrI4/FAs5Cs=">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</latexit>

(5, 13, 194)<latexit sha1_base64="8PT55cv4pTnfn+itTB9OvIUERDc=">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</latexit>(1, 5, 13)(1, 13, 34)<latexit sha1_base64="DXhko/6hK2PsPCEsOKO84FgXJy8=">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</latexit><latexit sha1_base64="CtoCF1cBVj4FKnboQhgF2d8mBQk=">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</latexit>(1, 1, 1)<latexit sha1_base64="kHE4k0S1NSp33POCFoKHE4wllt0=">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</latexit>(1, 1, 2)<latexit sha1_base64="0JNg2/WN2IqKk+bQa0HUWfmsUrs=">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</latexit>(1, 2, 5)(5, 29, 433)

<latexit sha1_base64="wyrE7XisU3OOjvQBqRIalqyC8x4=">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</latexit><latexit sha1_base64="V4H/Oqk/e7OnXd21fGanqjTGzcs=">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</latexit>(2, 5, 29)<latexit sha1_base64="Nv4MIXilLq9n/4lozWDGXkKMTsU=">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</latexit>(2, 29, 169)
<latexit sha1_base64="i44ar0nuOrMp/ca579SP55rbZhY=">AAACsXichVHLTttAFD0xtKXpg9BKVSU2qAFEJRSNjURJV1G7YZlAw0MEpbYzpKPYY8ueBNI0P9Af6KIrIrFA9CvKhh9gwSegLkFiw4LriTeUoo7luWfO3HPnXF0n9ESsGDvLGCOjDx4+GnucffL02fPx3MSLtThoRy6vuoEXRBuOHXNPSF5VQnl8I4y47TseX3daH5P79Q6PYhHIT6ob8m3fbkqxI1xbEVXPvarJQMgGlyo7PWfNW8V5c7H4drqey7MC02vqLjBTkEe6ykHuN2poIICLNnxwSCjCHmzE9G3BhIWQuG30iIsICX3P0UeWtDEhpc8STUJfKK9GkWOP9l3Kbmi2T3yRlD6hGr2V6EKq6KJFe5NOWykr6Zw4ifWbLnnz6I9IN4UZdsoO2QU7YUfsnF1rB/+q1dM1kg66FJ2hlof18e+vV6/+q/IpDj2nqnsViWeFHSxpr4K8h5pJunCH+s7XHxer71dmerNswP6Q/312xo6pA9m5dA8qfOUnsjQ28+8h3QVrVsFcKFgVK1/6kA5wDJN4gzma0juUsIwyqvTuNwxwhF/GgrFpfDacYaqRSTUvcWsZrRvIcZwI</latexit>

あるマルコフの３つ組みに属する数をマルコフ数という。
<latexit sha1_base64="XQy6Uli5HSRGN/TegUgrGbny9tY=">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</latexit>

マルコフ数は無限にあって，小さい数から順に<latexit sha1_base64="SjTf8DAYZzS7LLpltNL7OedZYj4=">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</latexit>

1, 2, 5, 13, 29, 34, 89, 169, 194, 233, 433, 610, 985, 1325, 1597, · · ·<latexit sha1_base64="fdTeEGgSd+LuP6f1UAvpmIv2WqY=">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</latexit>

マルコフ数とフィボナッチ数列
f0 = 0, f1 = 1, f2 = 1, f3 = 2, f4 = 3, · · · をフィボナッチ数列<latexit sha1_base64="7k56IOID0YsrSscwYOEkonrfEMI=">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</latexit>

このとき，(1, f2n 1 , f2n+1 )はマルコフの 3 つ組
<latexit sha1_base64="fyOn22jesRXHe7jNxEPC7dIvJCY=">AAAC/3ichVI9axRRFD0Zv+IazZo0gk1wiSjqcmdSGKxCtLARkqybBHaXZWb2ZTNkvpiZrCbLFmn9AxZWCiIiiGBlnSZ/IEUaexFsIsRC0DNvp4lRfMved+9599x7Lnec2PfSTORgxDh1+szZc6PnSxfGLl4aL1+eWE6jzcRVdTfyo2TVsVPle6GqZ17mq9U4UXbg+GrF2bifv6/0VJJ6Ufg424pVK7C7obfmuXZGKCqbaCJEBI+2A0WboSQ1CWVdHv0a42sDN2DiNtbQRh8WM+4wHhxDbmnkJrNbEsln+SofZV9CzEhn8KBdrkhV9Jk66ZiFU0FxFqLyLgt1KMrFJoJClAsfNlL+GmxmISbWYnsbCT1PvyuKKJGb0st0HKJLb515Td4KT2mfMLuj0QFxE0JuQL/JbjkzZk0XG7RdRo0CDRnnWlLd1aU6n/+EvClMc9i3cih78k6+yE+t4W+1+rpGPsMWb2fIVXF7/NmV2tF/WQHvoeqC9U9Grjnjema11ny1sUbyKdwhv7f9/LB2b2m6f11ecV378lIOZJcThL3v7utFtfQCJS7O/HNNJ51lq2rOVK1FqzI3X6xwFFdxTX82dzGHh1hAnX338A1H+GHsGG+M98aHYaoxUnAmcewYn34Dvx2uuQ==</latexit>

22
2
2(1, fn + fn+1 , fn+1 + fn+2 )はマルコフの３つ組み。

A. A. Markov (1856-1922)

以上は，興味をもたれた方への宿題<latexit sha1_base64="6GnF6PmcfAqLcIbupjQw2TGfNTs=">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</latexit>

マルコフ数に関する予想 (G. フロベニウス, 1913)<latexit sha1_base64="AvCb+pxH9fZbKwvRmqbXMkaI2ac=">AAACr3ichVE9TxtBEH0+QgKGBIc0SDQoFiiVNXcpIKkQNKkiPmIbhAncHYuz8n3p7jAfFn2U1IiCBiylICnzD0LDH3Dhhh6ldCQaCub2rklIlD3dztu382bfaKzAkVFM1M1pfQ/6Hz4aGMwPDT9+MlJ4OlqJ/J3QFmXbd/xwxTIj4UhPlGMZO2IlCIXpWo6oWo355L7aFGEkfe9dvB+Iddese3Jb2mbMVIOu6Cd9p44IqHHUplX6/PbjZm+jUKQSqTVxH+gZKCJbC37hB2rYgg8bO3Ah4CFm7MBExN8adBgImFtHi7mQkVT3AofIszZiFKuzhzqjD5xX4yiwx/suZ28p9pD5V6x0GdX4rUQXcEUbDd7rfFrLWI/PiZNIvWmzN4f/kHUTmKQOnVOPLukbXdOtcvC3Wi1VI+lgn6OVakWwMfJpbPnmvyqXY+o5U/1TkXiOsY0Z5VWy90AxSRd2qm8eHPeWXy9NtqaozSPr0Bl16YI78Jq/7C+LYukEeR6b/ueQ7oOKUdJfloxFozg7lw1wAON4jhc8pWnM4g0WUOZ3D3CKc3zVdK2qvdc201Qtl2me4belyTurdaAO</latexit><latexit sha1_base64="lIIA9Jpy22jSxX2MOOp6Z+rHbxo=">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</latexit>c をマルコフ数とすると c を最大値とするマルコフの 3 つ組 (a, b, c) はただ一つしかない。
(a b c）<latexit sha1_base64="q/qPcAQShMhl7jPII7vSNXg8XWY=">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</latexit><latexit sha1_base64="Rrv2zlG7i0zhuqBrFJxfe6iN6p8=">AAACu3ichVK7ThtBFD0sSQDnYQcaJBorhojKmnUKokhICBpKXgYkjNDuMDgbZh/aXTuARUfFD1CkClKKkDL8AU1+gIJPQCmJlCYSnB2vIiWAmNXOPXNmzr3nasaNtJekQlz0WL2PHj/p6x8oPH32/EWx9HJwOQlbsVR1GeowXnWdRGkvUPXUS7VajWLl+K5WK+72TLa/0lZx4oXBUrobqXXfaQbeliedlFRYKmIco3DQgIZCGe5fJDF6PbBRqoiqMKN8G9g5qCAfc2HpjAk2EVLegs9EAVJizRIJvzXYqCEit44OuZjIM/sK+yhQmxClZh2gSfSe5xqMCjucP/L0pmH3ydsQ1PrEDVbLlBFzSmxzbnK1lrMB15mXxFSVdKf5x9SVMSbOxVdxJX6Ib+JS/DEe7srVMTmyHnYZ3a5WRRvFw+HF3w+qfMau61x1ryLznGILb41Xj94jw2RdyK6+vXd0tfhuYazzWhyLn/T/WVyIM3YQtH/JL/Nq4RMKvDj7/2u6DZZrVftNtTZfq0xN51fYjxG84qOwMYEpzGIOddY9wAm+49SatKT1wdLdo1ZPrhnCP8Nq3QAfI5vR</latexit>

第3章

円とフラクタル図形

フラクタル(fractal)は1975年に自己相似集合など複雑な図形を扱うためにB.マ
ンデルブロートによって提唱された幾
何学の概念。非整数次元をもつ図形をフラクタルと呼ぶことが多いが，整数次元をもちながらフラクタルと呼ぶに相応しい図形も存在する。

Benoit Mandelbrot (1924-2010)

My soul is spiraling in frozen fractals all around.
— Frozen(アナと雪の女王) Let It Go より—

フラクタル図形の例とそのハウスドルフ次元

コッホ(Koch)曲線<latexit sha1_base64="9k38vt4nB+x0q3x1IECysy8lcIw=">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</latexit>

dimH () = log3 4 = 1.26186 · · ·<latexit sha1_base64="4/Wov7iTeXXd5NetmFLHgpA7d+w=">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</latexit><latexit sha1_base64="4HvewK/tuNVY0a9xP9qYEggAtmY=">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</latexit><latexit sha1_base64="AIwg4eNo4Umoj2tkgMIeR4lcmWY=">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</latexit>

シェルピンスキー(Sierpinski)の三角形<latexit sha1_base64="9k38vt4nB+x0q3x1IECysy8lcIw=">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</latexit>

dimH (<latexit sha1_base64="4HvewK/tuNVY0a9xP9qYEggAtmY=">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</latexit>) = log2 3 = 1.58496 · · ·<latexit sha1_base64="4/Wov7iTeXXd5NetmFLHgpA7d+w=">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</latexit><latexit sha1_base64="kljMJP77duixNahYuzILC3brIPQ=">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</latexit>

アポロニアン・パッキング<latexit sha1_base64="9k38vt4nB+x0q3x1IECysy8lcIw=">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</latexit>

dimH (<latexit sha1_base64="4HvewK/tuNVY0a9xP9qYEggAtmY=">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</latexit>)<latexit sha1_base64="4/Wov7iTeXXd5NetmFLHgpA7d+w=">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</latexit><latexit sha1_base64="z87vhnZz+CMFTwOCzlC4duYbXk0=">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</latexit>

1.305688 (McMullen, 1998)

ショットキー群(SCHOTTKY GROUP)<latexit sha1_base64="D+ZYLSN3Tul15zXuwYtzov85uYE=">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</latexit>

A2<latexit sha1_base64="R67kvOfl6CZPyyRbHMFB6HU9xpQ=">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</latexit>

A1

S1<latexit sha1_base64="ERBV6bA0KXDM6A3k94TmnXF1iDM=">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</latexit><latexit sha1_base64="AAbBTvSefyOqvDprjQFtHWOX5ZU=">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</latexit><latexit sha1_base64="SKhfHNHwkZdX/68QHpc2cJvh+8Q=">AAACp3ichVE5TsNAFH2YPSwJ0CDRIBIQNNHYFCAqBA0daxYpRsh2hmDFm2wnASIuwAGgoGCRKBDHoOECFBwBUQaJhoLviRs28S3PvP/mv5k383XPMoOQsecOqbOru6e3rz8xMDg0nEyNjOYDt+YbPGe4lusXdS3glunwXGiGFi96Ptds3eIFvboarRfq3A9M19kJjzy+a2sVx9w3DS0kSs2oZbfhaL7vNjJ7qTTLMhGTP4EcgzTi2HBTD1BRhgsDNdjgcBAStqAhoK8EGQo84nbRJM4nZIp1jhMkSBsQCkXuoELogOpUmjkOaWxQdVmwJ8TLYKS1Cat0WqT0aE8DVRorlJVi1qE88hKIUw1yZ9Hvk24S0+yJ3bEWe2T37IV9CA+/7dUUe0R3OKJZb2u5t5c8Hd9+/1dl09x2Hav+VESeQ+xjUXg1ybsnmOgWRltfPz5vbS9tTTdn2A17Jf/X7Jk90A2c+ptxu8m3LpCgxsnf2/QT5JWsPJ9VNpX08krcwj5MYAqz9LYLWMYaNpCjcz2c4RJX0py0LuWlYrtU6og1Y/gSkvYJNgaaOg==</latexit>

S2<latexit sha1_base64="abqD7tbmg1lrULJh4y6UVfKGuxw=">AAACoHichVFNLwNBGH66vqq+iovERRRxambXgTg1XNyoKkKl2V2jJvYru9tSjT/g4og4kTiIn9GLP+DgJ4gjiYuDd6d78RXvZmee95n3mXlmXsOzRBAy9pRQ2to7OruS3ame3r7+gfTg0HrgVn2TF03Xcv1NQw+4JRxeDEVo8U3P57ptWHzDOFiM1jdq3A+E66yFdY/v2HrFEXvC1EOiihOFsjZRTmdYlskY+wnUGGQQx4qbbqKEXbgwUYUNDgchYQs6Avq2oUKDR9wOGsT5hIRc5zhBirQBoVDmDiqE9qmuRDPHEY2HVL0r2RPiVTDS2oRLdFqk9GhPEwc0VijbjlmH8shLIE81yZ1Fv0+6MUyyR3bHXtkDu2fP7EN6+G2vhtwjukOdZqOl5V554HSk8P6vyqa55TpW/amIPIfYw5z0Ksi7J5noFmZLXzs+fy3Mr042ptgNeyH/1+yJNekGTu3NvM3z1SukqHHq9zb9BOtaVp3Janktk1uIW5jEKMYxTW87ixyWsIIinStwhgtcKuPKkrKs5FulSiLWDONLKFufpjmWmQ==</latexit><latexit sha1_base64="3zDbI5FM3R61+pkpHgY4vmTkxHY=">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</latexit>

B1<latexit sha1_base64="RDenGAq4RNlP0XVbY3VmiV4Lp+M=">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</latexit>

B2

Si (i = 1, 2) は Ai の外部を Bi の内部に写す一次分数変Fricke, Robert and Felix Klein(1912), Vorlesungen über die Theorie der automorphen Functionen.
換（必ず存在するが一意的ではない）<latexit sha1_base64="+nbnWh80NEMrZZOv5KQC7c+8O+A=">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</latexit>

ショットキー群と収束指数恒等変換 I と<latexit sha1_base64="4Arl0wrAqdW6ltyRJZw8vxAN3Js=">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</latexit>

S S S ···S<latexit sha1_base64="Pn5DdK4tiyQGWBS9KUOPwO7Kxys=">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</latexit>

1
i1

2
i2

3
i3

n
in

ik = 1 または ik = 2, k = ±1, n = 1, 2, 3, ....
ik = ik+1 のときは k + k+1 = 0<latexit sha1_base64="Mp4C76XCcO2z88XUYgxpPWfsMHo=">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</latexit><latexit sha1_base64="UkLkDwyuSBqUUNk6mjSSNYrl0FE=">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</latexit>• Sk 1 は Sk の逆変換
<latexit sha1_base64="deboXV+OnYbiJ358SZ5oihV+YLI=">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</latexit>の形の一次分数変換全体がつくる群 G をショットキー
群という。
<latexit sha1_base64="Uev/CQNfLcVR1DmJLf2WA+PV5ss=">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</latexit>

G の元に番号をつけてG=

an z + bn
Tn (z) =: n = 0, 1, 2, 3, · · ·cn z + dnとする。ただし, an dn<latexit sha1_base64="rfjQ+1WGSCWbWHSNJrhuA5F3jA0=">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</latexit>

bn cn = 1, T0 = I.

級数

1
s
|c
|
n
n=1

が収束する s の集合と発散する s の集合との境目 (G) をG の収束指数という。
<latexit sha1_base64="TvAtKmt9uWUDHmBtG3WPR2Dv2T0=">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</latexit>

ショットキー群が生み出すフラクタル図形

ショットキー群が生み出すフラクタル図形

無限個の円板に含まれる点の集合 L(G) を G の極限集合という。
<latexit sha1_base64="egumY4o0fCUvcptQKq5fnrD5d9k=">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</latexit>

定理 (赤座, 1973)(G)
dimH L(G) =.
2<latexit sha1_base64="w7jNzu0XguQQlvg3tPjVLQ/Gxow=">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</latexit><latexit sha1_base64="+42MIWidFx4LDhBLMjyYyLNNteo=">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</latexit>

1
s
|c
|
n
n=1

赤座暢(1927-1983)
1951年10月 名古屋大学から金沢大学理学部助教授（函数論講座）に就任

経歴

1972年8月

同教授（函数方程式講座）に昇進

1983年2月

肺癌のために死去

第4章 π は円周率か？

<latexit sha1_base64="vobZF72ZxxYvb70ACO7eUYZUylA=">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</latexit>

リーマン面の例
G をショットキー群とし，リーマン球面から円板 A1 , B1 ,
A2 , B2 を取り除き，Ak の境界円と Bk の境界円を Sk (k =1, 2) を用いて貼り合わせて得られる曲面==<latexit sha1_base64="u7CWg7T4xE5K6QHSqOfPN9zaFYs=">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</latexit><latexit sha1_base64="4cCi4pEprblmhaP5jUInwdEkQvU=">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</latexit>

リーマン面
各点のまわりに（局所的な）複素数のパラーメータが導入されていて，その上で複素関数論がちゃんと議論できる曲面(C<latexit sha1_base64="R1fR+1i5aoYSZ5VzZdSbe2R8QtE=">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</latexit>{ }L(G))/G

リーマン面の同値２つのリーマン面 R1 , R2 は等角写像（接平面の円を円に写す向きを保つ写像）

f : R1R2が存在するとき，同じリーマン面であるとみなす。
<latexit sha1_base64="9jcnG8bfAVXqZVQfZ9J8oGH345g=">AAADX3ichVIxb9NQED7HQFMT2rQsIJaItCgDii7uAOpUlQUkCnGaNK6SENkvr6kbx7ZsN1Ciig0J2LpEFRNIHRA/owsbEhJDB5AYgTFILJXC+cUCQUE86927u3ff3XfvbHq2FYSIR1JCPnX6zERyUjmbOjc1nZ6ZXQvcbZ/xCnNt19dNI+C25fBKaIU21z2fG13T5lWzcyO6r/a4H1iuUw53PN7oGm3H2rCYEZLLTRtQBwdcsEi2gJMMQRnNYgsd/ILH+BGHAxMyMAclaEKBzqs/LRXm0O3fevJO2y++HSXZwb1gYKKjPcY92h1tH9fLM7iCe3c/YEvETIJC9WqwAYuUZZyxThXd2FLJaoCCt2t6NYfr+Aw3cWWUKqdQ/8UG27gsbh7hFsUkmuks5lGszEmlECtZiFfRTR9SmRYVZbAN3bhpBjYYENBXI1IqeORrQJ98PmmWuOewKxoISAuF7UCbtE2Ki9rg8IDkfYpuCe8u+QuAhO2SXqdqEdKjnAw6JNtk1WKvQ3bEJRBVGbGzafuEy8A8vsdXOMQ3+Bo/47Hg8LdcfZEj6mGHTnOM5V5z+umF1e//RXXpHLOOUf9ERJxDGuJ1wTX6dTzhibpgY3zv4WC4ulia71/Bl/iV+L/AIzykDpzeN3ag8dJzUGhwhT/HdFJZU/OFhbyqqdml5XiESbgElyFHb3sNluAmFKECTMpJd6SqpCc+yRPylJwehyakGHMeflvyxR/bcsxT</latexit>リーマン面のモジュライ空間
位相的に n 個の穴を開けた種数 g の閉曲面であるリーマン面を (g, n) 型
リーマン面という。
<latexit sha1_base64="EUnMpdEUgrGDgUH9t19h2cf1TrM=">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</latexit>(g, n) 型モジュライ空間 Mg,n =（2,3）型曲面<latexit sha1_base64="JzfA27bMuY0c9UcFbJP0e+TUviA=">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</latexit>(g, n) 型リーマン面（の同値類）全体がつくる空間<latexit sha1_base64="uLPWQwTaPtxCMnJl3BQ52F9+/jE=">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</latexit>

モジュライ空間 Mg,n はヴェイユ・ピーターソン計量と呼ばれる計量をもつ。(他にもいろいろな計量をもつ。)<latexit sha1_base64="XFP7Dvpvn8BBGyAhln1XN27jDgI=">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</latexit>

しかも，ヴェイユ・ピーターソン計量で測った Mg,n の体積は有限値。
<latexit sha1_base64="NvtPmALfiBnG86auF2VsdjRfyT0=">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</latexit>

(１つ穴あき）トーラスのモジュライ空間の一つのモデル<latexit sha1_base64="3OlODkBwZoEnWs5yXnpgSmRW36k=">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</latexit>

2

2

2

x + y + z = 3xyz (x, y, z > 0)<latexit sha1_base64="5s3IWEMqXvC9hqSCQLTuzYKcpU4=">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</latexit>

2.5

2.0

<latexit sha1_base64="463JDzwoObucUejpqsx+piYTxac=">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</latexit>

3xz = y<latexit sha1_base64="haQpjAiealrJc+R6+cN7OSyuVRc=">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</latexit>

3yz = x

1.5<latexit sha1_base64="tmHibZ2L0t0M2np8phn/LRkonrA=">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</latexit>

x=z

1.0
<latexit sha1_base64="f4cL2tHFKXmE5QuCEwl5yPCbR8I=">AAAConichVG7SgNBFD1ZXzG+ojaCjRgVqzAbC0UQRBuxMsYkQjbI7jrGxX2xu0aTkB+wsxJMpWAhfkYaf8DCTxDLCDYW3p1s4wtn2blnztxz58xczTUNP2DsOSb19Pb1D8QHE0PDI6NjyfGJgu+cejrP647pePua6nPTsHk+MAKT77seVy3N5EXtZDPcL1a55xuOvRfUXF621IptHBm6GhBVVEq1tbpSPkimWJqJMfMTyBFIIRo7TrINBYdwoOMUFjhsBIRNqPDpK0FGBi5xZTSI8wgZYp+jiQRpfUKBWNuoEDqmPIUixznNZ5R9KNgm8TIYaS3CCp0WKl2qqeOE5gqtShFr0zr04otTdXJn0u+Rbgbz7Indsw57ZA/shX0ID7/Vaoga4R1qFLWulrsHYxdTufd/VRbFrutI9aci9BzgCCvCq0HeXcGEt9C7+mr9qpNb3Z1vLLBb9kr+b9gza9MN7Oqbfpfluy0kqHHy9zb9BIVMWl5KZ7KZ1PpG1MI4pjGLRXrbZaxjCzvIC3eXuEZLmpO2payU66ZKsUgziS9DUj4BMHiYIQ==</latexit><latexit sha1_base64="VfWILdCRDjYXJK1xv4mZ7pB7qW0=">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</latexit>(1,1,1)y=z

3xy = z

1.0

1.5

2.0

2.5

曲面 x2 + y 2 + z 2 = 3xyz を xy 平面に射影<latexit sha1_base64="rMbSjZFCLAQE1Rkzd2vcElX/8vc=">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</latexit>

（一つ穴あき）トーラスのモジュライ空間のヴェイユ・
ピーターソン体積（面積）<latexit sha1_base64="rTfusmAkQ+S/6V6k7LS9/YqvWYs=">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</latexit>

2

VolW P (1, 1) =<latexit sha1_base64="o/Nzdaa0WtZkNJXyX0hi9WS6OLc=">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</latexit>

6(S. Wolpert, 1983)<latexit sha1_base64="eS8w9uTHRL6LHVXOxEXDnq0/pKs=">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</latexit>

(g, n) 型リーマン面のモジュライ空間のヴェイユ・ピー
ターソン体積<latexit sha1_base64="9aq/9FM/6ExDroGoSSGtnBQgJEo=">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</latexit>

定理 (M. Mirzakhani, 2007)<latexit sha1_base64="qUJc1gFkArckE+vtifG+MotmbtU=">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</latexit><latexit sha1_base64="GRmMH86oYguYdQ6OgI0NAPG9tFs=">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</latexit><latexit sha1_base64="c7oDYPeU5OlUtvP1HDF2RwZN4QM=">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</latexit>

VolW P (g, n) = Q(g, n)

モジュライ空間の次元

6g 6+2n

ここで，Q(g, n) はある有理数。
<latexit sha1_base64="uSJajPCEM4KA3c2D+eLUbOzei4k=">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</latexit>

注意：Q(g, n) は双曲距離についての長さが指定された
境界つきリーマン面も考え，それらのモジュライ空間のヴェイユ・ピーターソン体積の間に成り立つ漸化式を用いて計算できる。
<latexit sha1_base64="bb2h/pFGDCMyw/Ef31QA4gRs6Es=">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</latexit>

ミルザハニによる結果は n > 0 を仮定する。
<latexit sha1_base64="J5BvQIp2W5FRJ4tbw0xvwqOU7eg=">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</latexit>

マリアム・ミルザハニ (Maryam Mirzakhani, 1977-2017)
1977年 イラン・テヘランで生まれる
1994-95年 国際数学オリンピックで金メダル獲得
2004年

ハーヴァード大学で博士号取得

2008年

スタンフォード大学教授（31歳）

2014年

フィールズ賞受賞（女性として初）

2017年

乳がんのために死去（40歳）「低い位置に実っている果実は、無視するべきです。
それは少々大変なことですが」
彼女は続ける。「こうした姿勢がベストなのかは分かりません。
その過程で、散々自分が苦しむことになりますから」- WIRED, 2014年8月24日 -ユークリッド幾何と非ユークリッド幾何

楕円幾何

曲率
<latexit sha1_base64="46ItY2PqYLMrJnbJ+4VhghpUdKg=">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</latexit>

1

ユークリッド幾何

曲率<latexit sha1_base64="3VK7RUvarDyMCCKTEcrShSD7bxM=">AAACtXichVG7ThtBFD1sEkLMy4QmUhorBpTKmjUFUSqUNJQ8YkACC+2Ox2bEvrK7NiGW00bKD6RIFQQFipKfoMkPpOATIjpAoqHg7Hib8FBmtTPnnrnnzpm5buTpJBXiZMB68PDR4OOhJ4XhkdGx8eLE09UkbMdS1WTohfG66yTK04GqpTr11HoUK8d3PbXm7rzN9tc6Kk50GLxL9yJV951WoJtaOikpv3mWfEYJU9iEwnu0odFhLDC1VSyLijCjdBvYOSgjH4th8ZhFGgghWcZnuQApsQcHCb8N2KgiIldHl1xMpM2+Qg8FahOi1MQBWkTbzNvkqvCB8y6zG4btkbdpUfKUHjPaRhmxpsQO5xajjZwNGGdeEnOqpDuPf0xdCdPijzgS5+K3+CH+iivj4a5aXVMju8MeV7evVdHW+JdnK5f/Vflc+65z1b2KzHOKJl4Zr5reI8Nkt5B9fefj1/OV18vT3RmxL07p/7s4Ece8QdC5kIdLavkbCmycfbNNt8FqtWLPVqpL1fL8m7yFQ3iOF3jJt53DPBawiBrP/YQD/MQva86qWw2r2U+1BnLNJP4ZVngN2Mqb0g==</latexit>

0

双曲幾何

曲率
<latexit sha1_base64="EPLYyFj9Ew16b61OHH3bxLW39aY=">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</latexit>

1「円」はすべてに幾何において円（定点から等距離にある点の集合）である。(ただし，中心と半径は異なる）

半径 r の円の周長と直径との比
楕円幾何 (K 1)
ユークリッド幾何 (K
双曲幾何 (K1)<latexit sha1_base64="EI4qTJphvftGwLQUFtly7AM9iv0=">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</latexit>0)

r 1 sin r ·r 1 sinh r ·Ｂ６１２では，π は円周率という名では存在し得な
い。でも，何らかの形で π という数字は見つかっていただろう。この惑星で π がどのように発見されるのか想像してみるのも楽しいかも知れません。

今日のまとめi<latexit sha1_base64="QYQegt70H2nYkGLB4Nxnxjd6s/o=">AAACqnichVG5TsNAEH2YOxwJ0CDRRAQQEihah4JDQkLQUEIgHEogsp1NWOFLtglHlB/gByioQFAg/gIafoCCT0CUINFQMN644RJreeft23mzbzS6awo/YOypSWluaW1r7+iMdXX39MYTff0bvnPgGTxnOKbjbemaz01h81wgApNvuR7XLN3km/r+Uni/WeWeLxx7PTh2+Y6lVWxRFoYWEFVMxEf4bq3giqSoT6jzbKSYSLE0kyv5E6gRSCFaK07iDgWU4MDAASxw2AgIm9Dg05eHigxc4nZQI84jJOQ9Rx0x0vqEAnm2USG0R3kFihxHtB9SdkmydeJnSWkRKtBboc6ligb2aa/QKR+xNp1DJ7580yBvJv0e6ZIYZY/shr2yB3bLntmHdPBbrZqsEXZwTFFvaLlbjJ8Orr3/q7IoNjxHqj8VoecAZcxIr4K8u5IJuzAa+urJ2evaXHa0NsYu2Qv5v2BP7J46sKtvxvUqz54jRmNTvw/pJ9jIpNWpdGY1k1pYjAbYgSEMY5ymNI0FLGMFOTnJc1zhWplUssq2km+kKk2RZgBfllL6BHkNmlk=</latexit>

e +1=0指数関数 e<latexit sha1_base64="lYuw2hbHlxfkH1NCkWriKgY33mg=">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</latexit>

ix<latexit sha1_base64="QMl38Y7Rsp4rk5+Omfq3flZ3mJs=">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</latexit>

e = cos x + i sin x

オイラーの公式 in 博士が愛した数式<latexit sha1_base64="LA1SRZJxLw0PoBax46UyNE1tUMg=">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</latexit>x

オイラーの公式

1
1,の収束・発散
2
4
|c
|
|c
|
n
n
n=1
n=1

1
の収束・発散
s
|c
|
n
n=1

(G)
dimH L(G) =赤座の定理
2<latexit sha1_base64="OT7ELy6iFceX9+5uLjw3vV6cxOU=">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</latexit><latexit sha1_base64="jVml7VaEGJ8gtjgE0xPNsUMzUUs=">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</latexit><latexit sha1_base64="l7lO0znXNAHIHECGMmXz4PttrVk=">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</latexit>

VolW P ()

Vg (L1 , L2 , L3 ) = VolW P (<latexit sha1_base64="l7lO0znXNAHIHECGMmXz4PttrVk=">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</latexit><latexit sha1_base64="Zi48G29i0tTx41puF8h0o9LLoWE=">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</latexit>

Mirzakhani の定理<latexit sha1_base64="48VQ2UkDtirHwSrO06blgFOKP3E=">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</latexit>

考える対象を連続的に変化する量や現象の中で捉えることによってうまくいった例を紹介した。

)

私の話はこれで終わりです。
どうも，ありがとうございました。

➤ 冲方丁「天地明察」角川書店, 2009年（文庫版（上・下),2012年）➤ 遠山啓「数学入門（上）」岩波新書, 1959年➤ 小川洋子「博士の愛した数式」新潮社, 2003年（文庫版,2005年）➤ 谷崎潤一郎「痴人の愛」改造社，1947年 （新潮文庫他）➤ M. Aigner, Markov’s Theorem and 100 Years of the UniquenessConjecture, Springer, 2013

➤ 小林吹代「マルコフ方程式」技術評論社，2017年➤ B. マンデルブロ，田沢恭子訳「フラクタリスト

マンデルブロ自伝」

早川書房, 2013年➤ 「とびきりの想像力が，女性初のフィールズ賞数学者を生んだ：マリアム・ミルザハニ」, Wired 2014年8月24日（インターネット上で閲覧可）