学⽣企画︓⾃然の形 植物に⾒られるフィボナッチ数と⻩⾦⾓137.5 学 企画...
TRANSCRIPT
137.5°
学⽣企画︓⾃然の形 植物に⾒られるフィボナッチ数と⻩⾦⾓奈良⼥⼦⼤学 理学部 ⾃然科学考房
牧野、濵野、⻘柳、⼩俵、津⽥、吉野、松尾、⾦井
3.観測結果
u ひまわりのらせん構造
4.MATHEMATICAによるシミュレーション結果
らせんの数/向き 13/左 21/右 34/左
u 松ぼっくりのらせん構造
uひまわりの筒状花の並びにみられるらせんの規則 (理論)
“A better way to construct the sunflower head”Helmut Vogel, Math. Biosciences. 44(1979): 179-189
u 筒状花の並びのシミュレーション結果1
点の数︓ 1000個 点の数︓1000個
θ ︓ 137.5° θ ︓ 137.6°
u シミュレーション結果1をもとにらせんを描いた結果点の数︓ 1000個 点の半径︓0.75 Θ ︓ 137.5°
らせんの向き/数
右13 34 89
左8 21 55
らせんの⾓度が⼤きい順に 8 13 21 34 55 89
• θ = 137.5° の時それぞれの花が重なることなく付く• らせんの数はフィボナッチ数に従い、向きは左右交互に現れる
• 実際にひまわりを育ててつぼみの部分を撮影し、花のつき⽅を観測する
• 奈良公園で松ぼっくりを採集し、観測する
2018/7/25 種まき『波紋と螺旋とフィボナッチ』(学研メディカル秀潤社・近藤 滋著)を読み、植物にフィボナッチ数列が⾒られることを知った。例えば、ロマネスコという植物に⾒られる螺旋の数はフィボナッチ数になっている。その他、パイナップルや松ぼっくりの果実やキク科の植物の花がフィボナッチ数列に従っているという。今回は、⽐較的採集しやすいひまわりと松ぼっくりを⽤いることで、フィボナッチ数が実際に⾒られるかを調べた。
1.研究の背景と⽬的 2.研究の⽅法A.ひまわりと松ぼっくりに⾒られるフィボナッチ数
1つの個体から何種類もの螺旋構造が⾒られた
植物の葉のつき⽅は⼤きく分けて、互⽣と輪⽣の2種類がある。互⽣植物の葉序にはフィボナッチ数が⾒られ、葉の開度は⻩⾦⾓であるという主張が確かめる。
ツツジ
イヌマキ◆互⽣植物の葉序に⾒られるフィボナッチ数
セイタカアワダチソウ
4.互⽣植物の葉序に関する結果とまとめ互⽣植物の葉序はらせん構造をもち、0番と重なる葉の番号はフィボナッチ数になることを確認した。3/8 葉序の開度は確かに⻩⾦⾓(137.5°)に近くなっていることがわかった。個体を変えると同じ種類の植物にも 2/7 葉序や 2/5 葉序も⾒られた。特にフィボナッチ数にならない 2/7 葉序に関しては今後さらに詳しく調べるつもりである。
' : 点の半径( : 点の番号
奈良⼥⼦⼤学
所在地 ︓ 奈良県奈良市学部構成︓ ⽂学部・理学部・⽣活環境学部理学部 ︓ 数物科学科
化学⽣物環境学科
物理学コース数学コース数物連携コース
・⽇本で⼆校ある国⽴の⼥⼦⼤学の1つ・奈良公園に近いため、キャンパス内に⿅がいる
化学コース⽣物学コース環境コース
奈良⼥⼦⼤学理学部⾃然科学考房
• 奈良⼥⼦⼤学理学部数物科学科で⾏っている学⽣主体の研究活動や⾃主ゼミなどを⽀援する事業
<活動内容>• 読書会 ︓輪読形式の⾃主ゼミ• 学⽣企画︓学⽣が主体となって⾏う
研究プロジェクト例)和算、ロボット、⾳の可視化など
: @suubutsu
• 科学イベントの開催・出展学びに触れる
考える発信する
Academic
合計16個の種 ⇒ 8本開花
• らせんの数がフィボナッチ数になっている• らせんの向きはらせんの数が増えるとともに左右交互に現れる
番⽬の花は中⼼から の距離に、番⽬の花から⾓度 の位置に付く
この時、中⼼と1番⽬の花の距離を1とする
k + 1<latexit sha1_base64="BCSDJQ2otrZDjqJGb7hrn+lY4es=">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</latexit><latexit sha1_base64="BCSDJQ2otrZDjqJGb7hrn+lY4es=">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</latexit><latexit sha1_base64="BCSDJQ2otrZDjqJGb7hrn+lY4es=">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</latexit><latexit sha1_base64="BCSDJQ2otrZDjqJGb7hrn+lY4es=">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</latexit>
pk + 1
<latexit sha1_base64="0dIAzPOTCwGGReOuCIdYYkUx5w0=">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</latexit><latexit sha1_base64="0dIAzPOTCwGGReOuCIdYYkUx5w0=">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</latexit><latexit sha1_base64="0dIAzPOTCwGGReOuCIdYYkUx5w0=">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</latexit><latexit sha1_base64="0dIAzPOTCwGGReOuCIdYYkUx5w0=">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</latexit>
k<latexit sha1_base64="AgrTKOTx7+zYFlu9MBJT5bqSNJY=">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</latexit><latexit sha1_base64="AgrTKOTx7+zYFlu9MBJT5bqSNJY=">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</latexit><latexit sha1_base64="AgrTKOTx7+zYFlu9MBJT5bqSNJY=">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</latexit><latexit sha1_base64="bs8QlLljyYde7WbNFuyzJJm+ZMk=">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</latexit><latexit sha1_base64="6O7732qj2mbxkWW2sLRDv+H7yJ0=">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</latexit><latexit sha1_base64="6O7732qj2mbxkWW2sLRDv+H7yJ0=">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</latexit><latexit sha1_base64="sMX7L6Qyw6An4e6PhYo0I111e9o=">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</latexit><latexit sha1_base64="AgrTKOTx7+zYFlu9MBJT5bqSNJY=">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</latexit><latexit sha1_base64="AgrTKOTx7+zYFlu9MBJT5bqSNJY=">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</latexit><latexit sha1_base64="AgrTKOTx7+zYFlu9MBJT5bqSNJY=">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</latexit><latexit sha1_base64="AgrTKOTx7+zYFlu9MBJT5bqSNJY=">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</latexit><latexit sha1_base64="AgrTKOTx7+zYFlu9MBJT5bqSNJY=">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</latexit><latexit sha1_base64="AgrTKOTx7+zYFlu9MBJT5bqSNJY=">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</latexit> ✓ = 137.5�
<latexit sha1_base64="HZc0HIaV8wtyWvEC0P+cetrRBbk=">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</latexit><latexit sha1_base64="HZc0HIaV8wtyWvEC0P+cetrRBbk=">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</latexit><latexit sha1_base64="HZc0HIaV8wtyWvEC0P+cetrRBbk=">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</latexit><latexit sha1_base64="bs8QlLljyYde7WbNFuyzJJm+ZMk=">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</latexit><latexit sha1_base64="kXQ4oSbpqux5pZrTbXx+vJnTwx4=">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</latexit><latexit sha1_base64="kXQ4oSbpqux5pZrTbXx+vJnTwx4=">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</latexit><latexit sha1_base64="hnErFNKe9tUxrOLuvfUAjVySnQg=">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</latexit><latexit sha1_base64="HZc0HIaV8wtyWvEC0P+cetrRBbk=">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</latexit><latexit sha1_base64="HZc0HIaV8wtyWvEC0P+cetrRBbk=">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</latexit><latexit sha1_base64="HZc0HIaV8wtyWvEC0P+cetrRBbk=">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</latexit><latexit sha1_base64="HZc0HIaV8wtyWvEC0P+cetrRBbk=">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</latexit><latexit sha1_base64="HZc0HIaV8wtyWvEC0P+cetrRBbk=">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</latexit><latexit sha1_base64="bs8QlLljyYde7WbNFuyzJJm+ZMk=">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</latexit><latexit sha1_base64="kXQ4oSbpqux5pZrTbXx+vJnTwx4=">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</latexit><latexit sha1_base64="kXQ4oSbpqux5pZrTbXx+vJnTwx4=">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</latexit><latexit sha1_base64="hnErFNKe9tUxrOLuvfUAjVySnQg=">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</latexit><latexit sha1_base64="HZc0HIaV8wtyWvEC0P+cetrRBbk=">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</latexit><latexit sha1_base64="HZc0HIaV8wtyWvEC0P+cetrRBbk=">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</latexit><latexit sha1_base64="HZc0HIaV8wtyWvEC0P+cetrRBbk=">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</latexit><latexit sha1_base64="HZc0HIaV8wtyWvEC0P+cetrRBbk=">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</latexit><latexit sha1_base64="HZc0HIaV8wtyWvEC0P+cetrRBbk=">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</latexit><latexit sha1_base64="HZc0HIaV8wtyWvEC0P+cetrRBbk=">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</latexit>
1.研究の背景と⽬的
奈良⼥⼦⼤学構内や周辺で種々の植物の観測を⾏った。また、セイタカアワダチソウ、ツツジ、イヌマキを採集して写真を撮り、開度を測った。
2 .研究の⽅法
B.互⽣植物の葉序に⾒られるフィボナッチ数
[参考⽂献] 岡部拓也 ⽣物科学 69 p.140-153 (2018)
任意の葉を基準(0番)とし、葉に番号をつけていくと
0番と重なるまでにした周回数0番と重なる葉の番号
は
1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34, 21/55などがよく⾒られる。ここに現れる数字がフィボナッチ数になっている。
3 .開度の結果葉の開度の平均値
ツ…ツツジ, セ…セイタカアワダチソウ, イ…イヌマキ本研究では様々な植物の葉序を観測し、その中でも 3/8 葉序がみられたセイタカアワダチソウ、イヌマキ、ツツジに注⽬してその開度を測定した。
ツツジ3には2/7葉序が⾒られたそれ以外の個体は全て3/8葉序が⾒られた
uフィボナッチ数列の漸化式
)* = )*,- +)*,/)- = 1 , )/ = 1
⇒ 1 1 2 3 5 8 13 21 ・・・
1+1
lim*→4
)*5-)*
= 1 + 52
⻩⾦⽐になっている︕
1 1 + 52
:
⻩⾦⽐
⻩⾦⾓
⾃然科学考房
(葉序の0番から6番までの開度を測定)
フィボナッチ数とは ⻩⾦⽐と⻩⾦⾓
[図・引⽤] ベネッセ教育情報サイト「⾃由研究 葉のつき⽅を調べよう︕」( )より
らせんの⾓度が⻩⾦⾓になっている⻩⾦⻑⽅形内の螺旋構造に似ている
例)オウムガイ
0番と1番の開度
ツ1
ツ2
ツ3
セ1
セ2
セ3
イ1
イ2
イ3
ツ4
⻑辺と短辺の⽐が1 1 + 5
2:
になっている⻑⽅形
⻩⾦⻑⽅形
1
1 + 52
正五⾓形
辺と対⾓線の⻑さが⻩⾦⽐になっている
8本 13本
360°を
に分割