Adinkra Symbol Mathematics

7/21/2019 Adinkra Symbol Mathematics

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Written by -K- ([email protected])www.theakan.comApril 2010

A Mathematical Analysis o Akan Adinkra !ymmetry
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2

"ontents

#ntrod$ction %

&asic 'ro$p heory and the ihedral 'ro$p *

-&rie !ketch o 'ro$p heory *

-he ihedral 'ro$p +

he ihedral ,ro$p and Adinkra symbol symmetries

- he Method

- /planation o es$lts

Where Mathematics meets Metaphysics 10

"oncl$sion 11

eerences 12

Appendi/ A /amples o Adinkra !ymbol !ymmetries 1%

Appendi/ & /amples o Adinkra !ymbols 13

Appendi/ " Adinkra !ymmetry etails 1+


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%

Introduction

he Akan 4Adinkra4 symbols are picto,rams and ideo,rams which ha5e come

to be well known as c$lt$ral-lin,$istic symbols o the Akan people o 'hana.

here are h$ndreds o these symbols. hey represent ideas s$ch as the

nat$re o the $ni5erse6 political belies and or,ani7ation6 social6 economic and

ethical 5al$es6 aesthetics as well as ideas relatin, to amily (see '. 8. K.

Arth$r4s book Cloth as Metaphor). hese symbols not only represent speciic

ideas and pro5erbs b$t are also related to other orms o Akan art s$ch as

brick layin, (b$ildin, ho$ses) metal crats6 wea5in, and wood car5in,6 all o

which can display 5ersions o these symbols in their work.

As to the ori,in o these symbols6 there has so ar not been a concl$si5e ordeiniti5e answer. !ome ha5e s$,,ested that the symbols deri5e rom #slamic

talismans. 9thers ha5e pointed o$t that this cannot be the case beca$se o

the block-printin, techni:$e $sed by the Akan (which has not chan,ed in

cent$ries) is dierent rom the writin, br$sh and stick method $sed in

#slamic inscriptions ('. 8. K. Arth$r6 20016 p22-23). What is known or s$re

is that the &ron6 the 'yaman and the enkyira6 three o the old centers o

Akan c$lt$re6 were in5ol5ed with wea5in, beore the Asante rose to power. #t

is also said that the name 4Adinkra4 comes rom a 'yaman kin, who wore the

cloth. 8or more inormation on the ori,in o these symbols6 see '. 8. K.

Arth$r4s book.

While attray p$blished 31 o these symbols in his book Religion and Art in

Ashanti6 '. 8. K Arth$r4s book contains ;1 o these symbols6 altho$,h some

are 5ariations o the same. 9$t o these h$ndreds o symbols there are many

with interestin, symmetrical properties that # became interested in st$dyin,.

Also6 $nlike attray4s 31 symbols6 some o the ;1 symbols ,i5en by '. 8. K.

Arth$r contain ideas rom modern times.

#n this paper6 # look at the symmetrical properties o some Adinkra symbols.

Altho$,h not all o these symbols ha5e symmetry6 there is a s$bset o them

that do possess symmetries that it a well st$died ,ro$p known as the4ihedral ,ro$p4 that describes symmetries o$nd in art as well as in some

,eometric shapes in nat$re. We shall be lookin, at this some more6 in the

ollowin, pa,es.


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*

Basic Group Theory and The Dihedral Group

i) Brief sketch of Group Theory

'ro$p theory is a branch o mathematics that looks at mathematical

str$ct$res with certain speciic properties. Most common ,ro$ps are inite

str$ct$res that ha5e internal consistency. 8or a mathematical str$ct$re to be

classiied as a ,ro$p6 it has to satisy certain conditions. here is an

operation which when perormed on one member o the ,ro$p ,i5es a res$lt

that points to another member o the ,ro$p. here are also the concepts o an

4identity46 the presence o an 4in5erse4 or each element and another concept in

mathematics called 4associati5ity4.

here are many6 many dierent kinds o str$ct$res that can be classiied as

,ro$ps6 b$t as a simple e/ample let $s $se the n$mbers called 4inte,ers46

which most people are amiliar with. he inte,ers are n$mbers like ... -%6 -26

-16 06 16 26 %6 ... he dots beore and ater the list o n$mbers says that the list

o n$mbers ,oes on ore5er in each direction6 positi5e and ne,ati5e.

As ar as the inte,ers are concerned6 the 4identity4 is the n$mber 7ero. What

this means is that any n$mber added to 7ero will ret$rn that n$mber. !o 1 -3 < 0 = -3> + < 0 = + etc. All that the identity does is ret$rn the

ori,inal element when the ,ro$p operation is applied. #n the case o inte,ers6

the ,ro$p operation is addition so when yo$ take any other element o the

,ro$p o inte,ers to,ether with the identity (7ero) and yo$ apply the ,ro$p

operation (addition) to these two elements yo$ ,et the non-identity element

(i.e. the other n$mber6 not 7ero). hat is what the e/amples abo5e showed.

he in5erse o an element in a ,ro$p is any other element in the ,ro$p which6

when the ,ro$p operation is applied to both elements ,i5es the identity. What

does this mean?

et $s ret$rn to o$r ,ro$p e/ample6 the inte,ers et $s choose any n$mber6

say +. What we want is to choose another n$mber so that when that n$mberis added to + will ,i5e $s 7ero. Bes6 yo$ ,ot it What we need is to choose -+6

so that + < (-+) = 0. #n this case6 we say that or o$r ,ro$p (inte,ers) with

,ro$p operation addition6 the in5erse o the n$mber + is -+ and the in5erse o

the n$mber -+ is +6 beca$se when both n$mbers are added to,ether6 we ,et

the 4identity46 which

*


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we ha5e already said is....7ero.

he inal re:$irement or a ,ro$p is this idea called 4associati5ity4. All that

this means can be demonstrated with an e/ample

1 < (2 < %) = (1 < 2) < % = +

!omeone who looks at this e/ample may say that we ha5e not done anythin,.

&$t this is incorrect. We $se MA! (some people learnt this as 4&9MA!4)

to e5al$ate the abo5e e/press. 4MA!4 stands or 4&rackets6 #nde/6

i5ision6 M$ltiplication6 Addition6 !$btraction46 while in &9MA!6 the 494

stands or 4order46 as in6 what somethin, is the power o. !o the two terms are

essentially the same. hey show which operations to work beore which.

What the e/ample abo5e shows is that the arithmetic operation 4addition4satisies this condition called 4associati5ity4 when applied to inte,ers. his

does not always work with e5ery arithmetic operation. et $s test this with

the same n$mbers

2 / (% / *) = (2 / %) / * = 2*

!o we can say that the arithmetic operation 4m$ltiplication4 is also

4associati5e4 when applied to inte,ers. What abo$t s$btraction?

2 - ( % C *) = % which is not e:$al to (2 C %) C * = -3

!o we can say that the arithmetic operation 4s$btraction4 is not associati5e

when applied to inte,ers. he same can be said o the arithmetic operation

4di5ision46 which shall now be shown

2 D (% D *) ,i5es 2.+++ (rec$rrin,) while (2 D %) D * ,i5es 0.1+++ (rec$rrin,) so

the two e/pressions are dierent dependin, on where the brackets are

placed.

!o associati5ity means Ewithin an e/pression containin, two or more

occ$rrences in a row o the same associati5e operator6 the order in which theoperations are perormed does not matter as lon, as the se:$ence o the

operands is not chan,ed.F (Wikipedia)

hat ends this brie treatment o basic ,ro$p theory. Gow let $s relate ,ro$p

theory ideas to the ihedral.


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+

ii) The Dihedral Group

he dihedral ,ro$p is the ,ro$p o symmetries o a re,$lar poly,on that

incl$de relections and rotations. he dihedral ,ro$p is a type o 4symmetry

,ro$p4.

# yo$ consider an e:$ilateral trian,le (trian,le with all three sides o e:$al

len,th6 and all three an,les o e:$al si7e)6 and i yo$ were to label the three

corners A6 &6 and "6 yo$ can ima,ine rotatin, the trian,le clockwise so that A

takes the position o &6 & takes the position o " and " takes the position o

A. his can be seen as one rotation. he an,le o rotation will be +0 de,rees6

since each an,le in an e:$ilateral trian,le has that an,le.

-------H

8i, 1 C otatin, a trian,le +0oclockwise

ikewise6 one can see how the trian,le can be rotated a,ain and a,ain so that

the corner labelled A will ret$rn to its ori,inal position. # one were to co$nt

the n$mber o rotations needed to do this6 the answer will be % (yo$ can try it

i yo$ like).

8i, 2 C ines o symmetry o a trian,le9ne can also draw lines o symmetry as shown in the i,$re abo5e. he


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trian,le has three o them6 while the s:$are has o$r o them. # we were to

carry o$t the symmetry operations on a s:$are whose * corners are labelled

(A6 &6 " and ) and which has * lines o symmetry6 we will prod$ce the table

o operations below

0 0 10 2;0 I J 4

0 0 0 10 2;0 I J 4

0 0 10 2;0 0 4 J I

10 10 2;0 0 0 J I 4

2;0 2;0 0 0 10 4 I

I I 4 J 0 10 2;0 0

J J I 4 10 0 0 2;0

I 4 J 0 2;0 0 10

4 4 J I 2;0 0 10 0

(!o$rce "ontemporary Abstract Al,ebra6 p. 23)

8i, % C 6 the symmetry ,ro$p o a s:$are

&asically this table says that 0is the ori,inal position o the s:$are6 so we

ha5e done nothin, to it. 0is a s:$are that has been rotated 0 de,reesclockwise. I is the hori7ontal line o symmetry o the s:$are6 J is the

5ertical6 is one dia,onal and 4 is the other dia,onal.

he reason why these operations orm a ,ro$p is that i we take the s:$are

$nder one o the operations and apply another operation to it the res$lt will

be a coni,$ration ittin, one o the ei,ht operations. !o all the operations are

inite6 closed and contained.

h$s the ihedral ,ro$p or the symmetries o a trian,le is called +6 beca$se

there are % rotations and % lines o symmetry. hat ,i5es + possible

operations. he dihedral ,ro$p or the symmetries o a s:$are is called 6 soyo$ may ha5e ,$essed where this is all leadin, to. We can state (witho$t

proo) that the dihedral ,ro$p o a re,$lar poly,on with n sides is called 2n.


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The Dihedral group and Adinkra symbol symmetries

i)The Method

here are many cases in nat$re where the symmetrical str$ct$re o an obect

are identical to a dihedral ,ro$p (o order n) or to a s$b,ro$p o a dihedral

,ro$p. his is the case with crystalline obects as well as with certain 5ir$ses

and protein str$ct$res. he e/ample ,i5en in Appendi/ A is a o a snowlake

that has the same symmetries as the dihedral ,ro$p 12(i.e. the symmetries

o a re,$lar he/a,on).

#t is :$ite well known that symmetrical art oten shows properties that can

be described by a dihedral ,ro$p. his is the main reason why # chose toe/amine the Adinkra symbols to ind o$t i there are partic$lar symmetries

that are more common than others.

#n order to do this6 # e/amined the inde/ o '. 8. K Arth$r4s book Cloth as

Metaphor(p 12 C 1;) where he lists ;1 Adinkra symbols6 to,ether with

their meanin,s6 pro5erbs and other associations. his list is perhaps the most

comprehensi5e list o Adinkra symbols in print (to date). #n decidin, which

Adinkra symbols to choose or analysis6 # (s$becti5ely) chose those that

looked $ni:$e since some o the symbols in the inde/ are 5ariations o the

same. 9$t o the ;1 symbols # sampled * o them6 which prod$ced a sample

o appro/imately 11.+L o all the symbols listed. he res$lt o the analysis is

shown in the table below6 while the more e/tensi5e list can be o$nd in

Appendi/ ".

Refl/Rot 0 ! " #"

0 2 1

$ 1*

% 1

! 3

"

#" 3

8i, * C Adinkra symmetries. !$mmary o sample o * symbols

ookin, at Appendi/ A will ,i5e some idea o how some Adinkra symbols can

be analysed to obtain their symmetries.


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i) %&planation of Results

he table in i,$re * (abo5e) shows the relection and rotational symmetries

o each symbol analysed (see Appendi/ A or how the symbols were analysed).

What i,$re * shows is that there are many dierent kinds o Adinkra symbol

symmetries. !ome shapes ha5e no relection or rotational symmetries while

others ha5e (potentially) ininite relection and (potentially) ininite

rotational symmetries (see n$mbers 3 and 2* in Appendi/ " or e/amples o

these).

8rom the sample o analysed symbols6 3 Adinkra symbols ha5e symmetries

that are identical to the dihedral ,ro$p +(symmetries o a trian,le) and

symbols ha5e symmetries identical to the dihedral ,ro$p . 3 symbols hadsymmetries that had more than * lines o symmetry and more than *

rotational symmetries (see n$mbers 6 20 and +0 in Appendi/ "). hese had

symmetries identical to dihedral ,ro$ps 1+and %2.

here were also shapes that had only one line o relection (i.e either

hori7ontal or 5ertical) or those that had 2 rotational symmetries. Appendi/ "

has more e/amples o these.

&$t by ar the most common were symbols that had 2 lines o symmetry

(hori7ontal and 5ertical) and two rotational symmetries. !hapes o this

nat$re can be seen as Eone i,$re made $p o two identical partsF. he

identical parts are mirror ima,es o each other. # think this is 5ery

interestin, i yo$ think in terms o an ima,e and its shadow or its mirror.

All the Adinkra symbols that it this 4most common pattern4 all $nder

dihedral ,ro$p *. hey are a s$b,ro$p o *witho$t all the transormations

(i.e. no dia,onal relections and two less rotations).

8inally it sho$ld perhaps be mentioned that that dihedral ,ro$p nis a

s$b,ro$p o dihedral ,ro$p 2n6 so that *is a s$b,ro$p o and is a

s$b,ro$p o 1+6 so that *is also a s$b,ro$p o 1+. !ince the irst re,$lar 2- shape we can ha5e is a a trian,le6 dihedral ,ro$ps start at +and contin$e

on to 6 10etc6 or re,$lar poly,ons o si7e %6 *6 36...


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10

'here Mathematics meets Metaphysics

M$ch as the mathematical symmetries o the Adinkra symbols are

interestin, $st by themsel5es6 # belie5e that at least some o the Adinkra

symbols encode 5ibrational ener,y patterns. his belie stems rom what #

ha5e obser5ed in some Gati5e #ndian sand paintin,s6 as well as Iind$ #ndian

yantras and ibetan mandalas. !ome o the Adinkra symbols are act$ally

(simpliied) carbon copies o patterns o$nd in some sand paintin,s6 mandalas

and yantras.

# belie5e that it may be possible to work o$t the relationship between the

e/ternal symmetries o some Adinkra symbols and their ener,etic-5ibrational

:$alities. Adinkra symbols may be associated with partic$lar colo$rs6n$mbers and so$nd patterns and hence may be $sed in a kabbalistic way.

#n order to in5esti,ate the 5ibrational ener,ies $nderlyin, some Adinkra

symbol symmetries6 # $sed an amethyst pend$l$m to dowse selected

Adinkra symbols. # obser5ed that o$r mo5ements maniested west-east6

north-so$th6 circ$lar (clockwise) and circ$lar (co$nter clockwise). 8$rther

work is bein, done in this re,ard to determine whether it is possible to come

$p with a schema showin, Adinkra symbols with 5ario$s adoinin, :$alities.

he symmetrical properties o these symbols may pro5ide a helpin, hand in

i,$rin, o$t this schema.

he a$thor o this article hopes that other spirit$ally oriented in5esti,ators

interested in Adinkra may cond$ct e/periments and in5esti,ations o their

own and share their res$lts. 9 partic$lar interest is what other indi5id$als

e/perience (so$nd6 colo$r6 mo5ement) when they meditate on or 5is$ali7e

partic$lar Adinkra symbols.


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(onclusion

Altho$,h it has emer,ed rom this in5esti,ation that the most common

symmetry or (symmetrical) Adinkra symbols is two lines o relection and

two rotations (to brin, the shape back to its ori,inal position)6 the story abo$t

the mathematical and possible metaphysical properties o Adinkra symbols is

not o5er. #n act it is only be,innin, #t is my hope that more in5esti,ations

will occ$r s$rro$ndin, these symbols in order to learn more abo$t them.


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R%)%R%*(%+

'. 8. K. Arth$r6 Cloth as Metaphor: (Re)-reading the Adinkra Cloth Symbols

of the Akan of Ghana6 A-oaks rintin, ress 1

N. A. 'allian6 Contemporary Abstract Algebra6 . ". Ieath and "ompany

10

. !. attray6 Religion and Art in Ashanti6 "larendon ress6 9/ord12;


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1%

Appendi& A , %&amples of Adinkra +ymbol +ymmetries


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Appendi& B , %&amples of Adinkra +ymbols

(!o$rce #nternet)

An Adinkra !ymbol6 a snowlake and a Ga5ao !and paintin,

(!o$rce #nternet)


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1+

Appendi& ( , Adinkra +ymmetry details

O eer to Adinkra #nde/ o book Cloth as Metaphorto see which

Adinkra symbols were $sed. he n$mbers in parentheses

correspond to the symbol n$mber in the inde/ o symbols.

*umber Adinkra *ame Rotational

+ymmetry

Reflection

+ymmetry

1 Abode !antann (1) 0 1

2 'ye Gyame (2) 2 0

% 9domankoma (1%) % %

* !oro ne Asase (1*) 2 0

3 $r$ (circle) (13) ininite ininite

+ Iann ne s$m (1) * *

; Asase ye d$r$ (236 2+) 2 2

Ananse Gtotan (%*)

!$ns$m (%3) * *

10 Gs$ (*0) 0 1

11 Iye Anhye (*2) 2 2

12 Gyame $a (3%) * *

1% Kerapa (+1) * 2

1* Momma ye mmbo mpae (;0) 2 2

13 Gkrabea (;3) 2 0

1+ 9w$o Atwede (0) 2 2

1; 9w$o Mpe !ika () 2 2

1 Mema wo hyeden (103) 2 2

1 Woye hwan? (111) 2 2

20 Gsoromma (11+) 21 &iribi wo soro (12+) 2 2

22 9hene Adwa (131) 0 1

2% !$mpie (1;*) % %

2* Adinkra hene (1;-13) otentially

ininite

otentially

ininte


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23 Aban pa (1%) 2 2

2+ Gk$r$ma kese (13) * *

2; wennimen (212) 2 2

2 sono Anantam (222) * *

2 Gkotimseo $a (2%*) * 0

%0 'yaw$ Atiko (2*;) 2 0

%1 Akoben (23%) 0 1

%2 kyem (2+) 0 1

%% 9bi nka obi (2;%) 2 0

%* wan ire (21) 0 1

%3 Mmara Krado (20) 2 2

%+ pa (2+) 2 2

%; !epo (2) 0 1

% Gkonsonkonson (%0*) 2 2

% se ne tekrema (%22) 2 2

*0 empasie (%2%) 2 2

*1 &oa me (%%1) 2 2

*2 &oa w4aban (%%+) 2 2

*% 8awohodie (%%) 2 2

** $a koro (%*%) 0 1

*3 Adwo (%*) 2 2

*+ Mpaboa (%3%) 2 2

*; Asase Aban (%3*) % %

* Asomdwoe 2 2

* Asambo (%;3) 2 2

30 Mmora &anyini (*21) 2 231 Akoko nan (*2*) 0 1

32 Ab$s$a do $n$ (*3+) 2 2

3% Be papa (*+%) 2 2

3* &$ wo ho (*+*) 2 2

33 Gni aw$ (*;) 0 0


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3+ Kata wo de so (*3) 2 2

3; nni nsek$ro (*%) 0 1

3 Wawa aba (30+) 2 2

3 Aya (31) 0 1

+0 W4ano pe asem (32*) 1+ 1+

+1 Aka m4eni (3%0) 0 1

+2 Mekyia wo (3%1) 2 0

+% !aa? (3%+) 2 2

+* Gantie yie (3*2) 2 2

+3 9bra ye bona (3*;) % %

++ Adasa pe mmoboro (3*) 0 1

+; Iwe yie (33*) 2 2

+ &o wo ho ban (333) 2 2

+ Iwe w4akan m$ yie (33+) 2 0

;0 Mens$ro wo (33) 0 0

;1 wen wo ho (3+%) 2 2

;2 &ese saka (3;+) * *

;% Asetena pa (+0*) 2 2

;* !ankoa (+*2) 0 1

;3 Mate Masie (+32) * *

;+ K$ntankantan (+++) * *

;; Gkore (+;0) 2 2

; 'yina intinn (+;2) 2 2

; wen hwe kan (+;%) 2 2

0 W$d$ nkwanta a (+%) 2 2

1 Ab$r$b$ro kos$a (;10) 2 02 Ananse anton kasa (;13) 2 2

% Ahinansa (;1;) % %

* Gkabom (;1) 2 0

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Adinkra Symbol Mathematics