對稱

對稱（粵拼：deoi3 cing3）係幾何學上，指圖形嘅某種重複性。常見有平移、旋轉、鏡射等。對稱嘅意思係當你折疊或者翻轉一個形狀嘅一部分嘅時候，佢睇起嚟同另一部分一模一樣。試想像喺一個心形中間劃一條線。如果兩邊完全匹配，噉呢個心形就有線對稱嘅特性。有啲形狀轉動或者旋轉之後仲係一樣樣嘅，呢種就叫旋轉對稱。對稱可以令到嘢睇起嚟平衡同勻稱，好似蝴蝶嘅翼或者雪花噉。

對稱唔一定係具象嘅，數學上，對稱性其實係講，當一樣嘢，佢嘅某啲方面改變，而另一啲方面維持唔變。群論就係研究對稱性嘅數學分支。

概論

嚴格嚟講，當我哋話一個數學物體係對稱嘅時候，即係話呢個物體就算經過反射或者轉動等變換，佢嘅樣都唔會改變。舉個例，鏡射係最基本嘅一種對稱，意思係話一個物體經過反射之後都保持原狀^[1]^[2]。講得簡單啲，反射可以咁樣理解：

攞一個形狀（好似下圖嘅三角形 $abc$ ）同一條線（就係圖入面嘅 Y 軸），呢條線就叫做「反射軸」；
喺呢條線嘅另一邊整一個新嘅形狀（三角形 $a'b'c'$ ）；
原本個形狀嘅每一點 $p_{0}$ ，喺新形狀度都有個對應嘅點 $p_{1}$ ；
隨便揀兩個對應嘅點，「 $p_{0}$ 同反射軸之間嘅距離」一定同「 $p_{1}$ 同反射軸之間嘅距離」一樣。

如果一件物體經過反射之後，佢反射前同反射後嘅樣完全一樣（除咗位置之外），咁就話呢件物體有鏡射對稱嘅特性。再深入啲嘅對稱分析，仲有講到轉動對稱（即係話一件物體轉動一定角度之後都唔會變樣，好似三曲腿圖咁，轉 120 度都係同一個樣）等更加複雜嘅對稱類型。

三角形

abc

沿 Y 軸反射，得出

a'b'c'

呢個新嘅三角形。

呢個梯形有左右鏡射對稱特性。

生物學

喺生物學入面，對稱嘅概念主要用嚟描述生物體嘅形狀。雙側動物（例如人類噉），喺將身體分成左右兩半嘅矢狀面上或多或少係對稱嘅。^[3] 向一個方向移動嘅動物必然有上下兩面、頭尾兩端，因此有左右之分。頭部變得專門化，有口同感覺器官，而身體為咗移動嘅目的變成雙側對稱，有對稱嘅肌肉同骨骼元素，不過內部器官通常仍然係唔對稱嘅。^[4]

植物同固着嘅動物，好似海葵，通常有輻射對稱或者旋轉對稱，呢個特徵好適合佢哋，因為食物或者威脅可能會從任何方向到來。五重對稱喺棘皮動物中可以搵到，呢個類群包括海星、海膽同海百合。^[5]

喺生物學入面，對稱嘅概念都好似物理學咁使用，即係用嚟描述所研究對象嘅特性，包括佢哋嘅相互作用。生物進化嘅一個顯著特性就係對稱性嘅改變，呢個對應住新嘅部分同動態嘅出現。^[6]^[7]

心理學

對於人類觀察者嚟講，有啲對稱類型比其他類型更加突出，特別係垂直軸嘅反射對稱最為顯著，就好似人類面部嘅對稱咁。恩斯特·馬赫喺佢嘅著作《感覺分析》（1897年）中就提出咗呢個觀察，^[8] 呢個表示對稱嘅感知唔係對所有類型嘅規律性嘅一般反應。行為同神經生理學研究都證實咗人類同其他動物對反射對稱嘅特殊敏感性。^[9] 早期喺格式塔傳統中嘅研究表明，雙側對稱係知覺分組嘅關鍵因素之一。呢個就係所謂嘅對稱法則。對稱喺分組同圖形/背景組織中嘅作用已經喺好多研究中得到證實。例如，當反射對稱係單一物體嘅特性時，檢測呢種對稱嘅速度會更快。^[10] 人類知覺同心理物理學嘅研究顯示，對稱嘅檢測係快速、有效同對擾動具有強大抵抗力嘅。例如，喺100到150毫秒之間嘅展示時間內就可以檢測到對稱。^[11]

美學

對稱喺建築設計嘅每個層面都有體現，由建築物外觀嘅整體設計，好似哥特式大教堂同白宮，到個別平面圖嘅佈局，再到建築元素嘅細節設計，例如瓷磚馬賽克。伊斯蘭教建築，好似泰姬陵同洛特福拉清真寺，喺結構同裝飾上都大量運用咗對稱性。^[12]^[13]。

摩爾人嘅建築，例如阿爾罕布拉宮，好興用複雜嘅圖案裝飾，呢啲圖案運用咗平移對稱、反射對稱同旋轉對稱。^[14]

睇埋

引咗

↑ Symmetry. Wolfram MathWorld.
↑ James, E. (2005). Symmetry and the Beautiful Universe. Journal of College Science Teaching, 34(5), 55.
↑ Valentine, James W. "Bilateria". AccessScience. 原著喺18 January 2008歸檔. 喺29 May 2013搵到.
↑ Hickman, Cleveland P.; Roberts, Larry S.; Larson, Allan (2002). "Animal Diversity (Third Edition)" (PDF). Chapter 8: Acoelomate Bilateral Animals. McGraw-Hill. p. 139. 原著 (PDF)喺May 17, 2016歸檔. 喺October 25, 2012搵到.
↑ Stewart, Ian (2001). What Shape is a Snowflake? Magical Numbers in Nature. Weidenfeld & Nicolson. pp. 64–65.
↑ Longo, Giuseppe; Montévil, Maël (2016). Perspectives on Organisms: Biological time, Symmetries and Singularities (英文). Springer. ISBN 978-3-662-51229-6.
↑ Montévil, Maël; Mossio, Matteo; Pocheville, Arnaud; Longo, Giuseppe (2016). "Theoretical principles for biology: Variation". Progress in Biophysics and Molecular Biology. From the Century of the Genome to the Century of the Organism: New Theoretical Approaches. 122 (1): 36–50. doi:10.1016/j.pbiomolbio.2016.08.005. PMID 27530930. S2CID 3671068.
↑ Mach, Ernst (1897). Symmetries and Group Theory in Particle Physics: An Introduction to Space-Time and Internal Symmetries. Open Court Publishing House.
↑ Wagemans, J. (1997). "Characteristics and models of human symmetry detection". Trends in Cognitive Sciences. 1 (9): 346–352. doi:10.1016/S1364-6613(97)01105-4. PMID 21223945. S2CID 2143353.
↑ Bertamini, M. (2010). "Sensitivity to reflection and translation is modulated by objectness". Perception. 39 (1): 27–40. doi:10.1068/p6393. PMID 20301844. S2CID 22451173.
↑ Barlow, H.B.; Reeves, B.C. (1979). "The versatility and absolute efficiency of detecting mirror symmetry in random dot displays". Vision Research. 19 (7): 783–793. doi:10.1016/0042-6989(79)90154-8. PMID 483597. S2CID 41530752.
↑ Williams: Symmetry in Architecture. Members.tripod.com (1998-12-31). Retrieved on 2013-04-16.
↑ Aslaksen: Mathematics in Art and Architecture. Math.nus.edu.sg. Retrieved on 2013-04-16.
↑ Derry, Gregory N. (2002). What Science Is and How It Works. Princeton University Press. pp. 269–. ISBN 978-1-4008-2311-6.

[1] Symmetry. Wolfram MathWorld.

[2] James, E. (2005). Symmetry and the Beautiful Universe. Journal of College Science Teaching, 34(5), 55.

[3] Valentine, James W. "Bilateria". AccessScience. 原著喺18 January 2008歸檔. 喺29 May 2013搵到.

[4] Hickman, Cleveland P.; Roberts, Larry S.; Larson, Allan (2002). "Animal Diversity (Third Edition)" (PDF). Chapter 8: Acoelomate Bilateral Animals. McGraw-Hill. p. 139. 原著 (PDF)喺May 17, 2016歸檔. 喺October 25, 2012搵到.

[5] Stewart, Ian (2001). What Shape is a Snowflake? Magical Numbers in Nature. Weidenfeld & Nicolson. pp. 64–65.

[6] Longo, Giuseppe; Montévil, Maël (2016). Perspectives on Organisms: Biological time, Symmetries and Singularities (英文). Springer. ISBN 978-3-662-51229-6.

[7] Montévil, Maël; Mossio, Matteo; Pocheville, Arnaud; Longo, Giuseppe (2016). "Theoretical principles for biology: Variation". Progress in Biophysics and Molecular Biology. From the Century of the Genome to the Century of the Organism: New Theoretical Approaches. 122 (1): 36–50. doi:10.1016/j.pbiomolbio.2016.08.005. PMID 27530930. S2CID 3671068.

[8] Mach, Ernst (1897). Symmetries and Group Theory in Particle Physics: An Introduction to Space-Time and Internal Symmetries. Open Court Publishing House.

[9] Wagemans, J. (1997). "Characteristics and models of human symmetry detection". Trends in Cognitive Sciences. 1 (9): 346–352. doi:10.1016/S1364-6613(97)01105-4. PMID 21223945. S2CID 2143353.

[10] Bertamini, M. (2010). "Sensitivity to reflection and translation is modulated by objectness". Perception. 39 (1): 27–40. doi:10.1068/p6393. PMID 20301844. S2CID 22451173.

[11] Barlow, H.B.; Reeves, B.C. (1979). "The versatility and absolute efficiency of detecting mirror symmetry in random dot displays". Vision Research. 19 (7): 783–793. doi:10.1016/0042-6989(79)90154-8. PMID 483597. S2CID 41530752.

[12] Williams: Symmetry in Architecture. Members.tripod.com (1998-12-31). Retrieved on 2013-04-16.

[13] Aslaksen: Mathematics in Art and Architecture. Math.nus.edu.sg. Retrieved on 2013-04-16.

[14] Derry, Gregory N. (2002). What Science Is and How It Works. Princeton University Press. pp. 269–. ISBN 978-1-4008-2311-6.

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