查看“︁吠陀方形”︁的源代码

__NOTOC__
'''[[吠陀]]方形'''（Vedic square）屬於古[[印度數學]]，是9&nbsp;&times;&nbsp;9 [[乘法表]]的變形，每個數字都用乘積的[[數根]]來代替。換句話說，與乘積除以9以後的[[余数]]的概念接近，若是該乘積為9的倍數，其[[數根]]為9不為0。 吠陀方形中有許多[[幾何]][[模式]]及[[對稱]]特性，其中有些模式會出現在傳統的[[伊斯蘭藝術]]<ref>{{Cite news|url=http://pansci.asia/archives/110479|title=這個九九乘法表你小學沒背過！吠陀方形的千年奧秘|last=|first=|date=2016-12-06|work=|newspaper=[[PanSci 泛科學]]|accessdate=2017-01-17|language=zh-TW|archive-date=2020-08-12|archive-url=https://web.archive.org/web/20200812062246/https://pansci.asia/archives/110479|dead-url=no}}</ref>。[[File:Shapes in the Vedic Square.png|thumb|標示吠陀方形中特定數字的位置，可以看出有某種[[軸對稱]]]]

<div align="center">
{|class="wikitable" style="text-align:center; width:270px; height:270px" border="1"
! <math>\circ</math> !! 1 !! 2 !! 3 !! 4 !! 5 !! 6 !! 7 !! 8 !! 9
|-
! 1
|style="background-color:#fbb"|1||style="background-color:#EFD7FF"|2||style="background-color:#ffa"|3||style="background-color:#EAFFEF"|4||style="background-color:#A4F0B7"|5||style="background-color:#BBDAFF"|6||style="background-color:#CEFFFD"|7||style="background-color:#EEEEFF"|8||style="background-color:#2FAACE"|9
|-
!2
|style="background-color:#EFD7FF"|2||style="background-color:#EAFFEF"|4||style="background-color:#BBDAFF"|6||style="background-color:#EEEEFF"|8||style="background-color:#fbb"|1||style="background-color:#ffa"|3||style="background-color:#A4F0B7"|5||style="background-color:#CEFFFD"|7||style="background-color:#2FAACE"|9
|-
!3
|style="background-color:#ffa"|3||style="background-color:#BBDAFF"|6||style="background-color:#2FAACE"|9||style="background-color:#ffa"|3||style="background-color:#BBDAFF"|6||style="background-color:#2FAACE"|9||style="background-color:#ffa"|3||style="background-color:#BBDAFF"|6||style="background-color:#2FAACE"|9
|-
!4
|style="background-color:#EAFFEF"|4||style="background-color:#EEEEFF"|8||style="background-color:#ffa"|3||style="background-color:#CEFFFD"|7||style="background-color:#EFD7FF"|2||style="background-color:#BBDAFF"|6||style="background-color:#fbb"|1||style="background-color:#A4F0B7"|5||style="background-color:#2FAACE"|9
|-
!5
|style="background-color:#A4F0B7"|5||style="background-color:#fbb"|1||style="background-color:#BBDAFF"|6||style="background-color:#EFD7FF"|2||style="background-color:#CEFFFD"|7||style="background-color:#ffa"|3||style="background-color:#EEEEFF"|8||style="background-color:#EAFFEF"|4||style="background-color:#2FAACE"|9
|-
!6
|style="background-color:#BBDAFF"|6||style="background-color:#ffa"|3||style="background-color:#2FAACE"|9||style="background-color:#BBDAFF"|6||style="background-color:#ffa"|3||style="background-color:#2FAACE"|9||style="background-color:#BBDAFF"|6||style="background-color:#ffa"|3||style="background-color:#2FAACE"|9
|-
!7
|style="background-color:#CEFFFD"|7||style="background-color:#A4F0B7"|5||style="background-color:#ffa"|3||style="background-color:#fbb"|1||style="background-color:#EEEEFF"|8||style="background-color:#BBDAFF"|6||style="background-color:#EAFFEF"|4||style="background-color:#EFD7FF"|2||style="background-color:#2FAACE"|9
|-
!8
|style="background-color:#EEEEFF"|8||style="background-color:#CEFFFD"|7||style="background-color:#BBDAFF"|6||style="background-color:#A4F0B7"|5||style="background-color:#EAFFEF"|4||style="background-color:#ffa"|3||style="background-color:#EFD7FF"|2||style="background-color:#fbb"|1||style="background-color:#2FAACE"|9
|-
!9
|style="background-color:#2FAACE"|9||style="background-color:#2FAACE"|9||style="background-color:#2FAACE"|9||style="background-color:#2FAACE"|9||style="background-color:#2FAACE"|9||style="background-color:#2FAACE"|9||style="background-color:#2FAACE"|9||style="background-color:#2FAACE"|9||style="background-color:#2FAACE"|9
|}
</div>

==代數性質==
吠陀方形可以視為是[[幺半群]]<math>((\mathbb{Z}/9\mathbb{Z})^{\times}, \{1, \circ\})</math>的[[乘法表]]，其中<math>\mathbb{Z}/9\mathbb{Z}</math>是整數除以9後所可能的[[餘數]]（運算元''<math>\circ</math>''是指幺半群元素之間的抽象乘法）

若<math>a,b</math>是<math>((\mathbb{Z}/9\mathbb{Z})^{\times}, \{1, \circ\})</math> 的元素，則<math>a \circ b</math>可以定義為<math>(a \times b) \mod{9}</math>，其中元素9表示其除以9以後餘數為0，而不用傳統的0來表示。

這個幺半群不是數學上的[[群]]，因為不是每一個非零元素都有對應的[[逆元素]]，例如<math>6\circ 3 = 9</math>，但不存在<math>a \in \{ 1,\cdots,9 \}</math>使得<math>9\circ a = 6.</math>。

===子集的性質===
子集<math>\{1,2,4,5,7,8\}</math>形成[[循環群]]<!--with 2 as one choice of [[群的生成集合|Generating set of a group|generator]] - this is the group of multiplicative [[单位 (环论)|Unit (ring theory)|units]] in the [[环 (代数)|Ring (mathematics)|ring]] <math>\mathbb{Z}/9\mathbb{Z}</math>. -->。每一行及每一列都恰好有六個相異的數字，因此這個子集也是[[拉丁方陣]]。

<div align="center">
{|class="wikitable" style="text-align:center; width:270px; height:270px" border="1"
! <math>\circ</math>  !! 1 !! 2 !! 4 !! 5 !! 7 !! 8
|-
! 1
|style="background-color:#fbb"|1||style="background-color:#EFD7FF"|2||style="background-color:#EAFFEF"|4||style="background-color:#A4F0B7"|5||style="background-color:#CEFFFD"|7||style="background-color:#EEEEFF"|8
|-
!2
|style="background-color:#EFD7FF"|2||style="background-color:#EAFFEF"|4||style="background-color:#EEEEFF"|8||style="background-color:#fbb"|1||style="background-color:#A4F0B7"|5||style="background-color:#CEFFFD"|7
|-
!4
|style="background-color:#EAFFEF"|4||style="background-color:#EEEEFF"|8||style="background-color:#CEFFFD"|7||style="background-color:#EFD7FF"|2||style="background-color:#fbb"|1||style="background-color:#A4F0B7"|5
|-
!5
|style="background-color:#A4F0B7"|5||style="background-color:#fbb"|1||style="background-color:#EFD7FF"|2||style="background-color:#CEFFFD"|7||style="background-color:#EEEEFF"|8||style="background-color:#EAFFEF"|4
|-
!7
|style="background-color:#CEFFFD"|7||style="background-color:#A4F0B7"|5||style="background-color:#fbb"|1||style="background-color:#EEEEFF"|8||style="background-color:#EAFFEF"|4||style="background-color:#EFD7FF"|2
|-
!8
|style="background-color:#EEEEFF"|8||style="background-color:#CEFFFD"|7||style="background-color:#A4F0B7"|5||style="background-color:#EAFFEF"|4||style="background-color:#EFD7FF"|2||style="background-color:#fbb"|1
|}
</div>

== 三維的吠陀立方==
吠陀立方定義為三維[[乘法表]]中，用每個乘積的[[數根]]來代替乘積<ref>{{Cite web|url=http://rmm.ludus-opuscula.org/Home/ArticleDetails/1155|title=Digital root patterns of three-dimensional space|accessdate=2017-01-18|author=Chia-Yu Lin|date=|publisher=|website=rmm.ludus-opuscula.org|access-date=|archive-date=2020-02-08|archive-url=https://web.archive.org/web/20200208183225/http://rmm.ludus-opuscula.org/Home/ArticleDetails/1155|dead-url=no}}</ref><ref>{{Cite news|url=http://pansci.asia/archives/112219|title=數字感有什麼用？他把風靡千年的吠陀方形變立體了！|last=|first=|date=2016-12-31|work=|newspaper=[[PanSci 泛科學]]|accessdate=2017-01-17|language=zh-TW|archive-date=2020-10-01|archive-url=https://web.archive.org/web/20201001200116/https://pansci.asia/archives/112219|dead-url=no}}</ref>。

==相關條目==
*[[拉丁方陣]]
*[[模運算]]
*[[幺半群]]

==參考資料==
{{Reflist}}
*{{citation|last=Deskins|first=W.E.|title=Abstract Algebra|publisher=Dover|location=New York|pages=162–167|isbn=0-486-68888-7|year=1996}}
*{{citation|last=Pritchard|first=Chris|title=The Changing Shape of Geometry: Celebrating a Century of Geometry and Geometry Teaching|publisher=Cambridge University Press|location=Great Britain|pages=119–122|isbn=0-521-53162-4|year=2003}}
*{{citation|last=Ghannam|first=Talal|year=2012|title=The Mystery of Numbers: Revealed Through Their Digital Root|publisher=CreateSpace Publications|pages=68–73|isbn=978-1-4776-7841-1}}
*{{citation|last=Teknomo|first=Kadi|year=2005|url=http://people.revoledu.com/kardi/tutorial/DigitSum/Vedic-square.html|title=Digital Root: Vedic Square|accessdate=2017-01-17|archive-date=2019-10-29|archive-url=https://web.archive.org/web/20191029112509/https://people.revoledu.com/kardi/tutorial/DigitSum/Vedic-square.html|dead-url=no}}
*{{citation|last=Chia-Yu|first=Lin|year=2016|url=http://rmm.ludus-opuscula.org/Home/ArticleDetails/1155|title=Digital Root Patterns of Three-Dimensional Space|publisher=Recreational Mathematics Magazine|pages=9–31|issn=2182-1976|accessdate=2017-01-17|archive-date=2020-02-08|archive-url=https://web.archive.org/web/20200208183225/http://rmm.ludus-opuscula.org/Home/ArticleDetails/1155|dead-url=no}}

<!--{{Indian mathematics}}-->
<!--[[Category:Indian mathematics]]-->
[[Category:同余]]