Transcript Group Theory and the Warlpiri Tribe

Group Theory and the Warlpiri
Tribe:
Sociological setup can be
represented by D8 group and
proved a
deterrent of incestuous
relationships
Natalie Skall
Saint Michael’s College
Colchester, VT
Spring 2004
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Kin Relations
Mathematic concept of relations extended to people
Analyze properties, such as symmetry, transitivity,
composite relations
Symmetry: b is the brother of c; is c the brother of b?
Transitivity: b is the brother of c, c is the brother of d; is b
the brother of d?
Composite relations: the mother of the mother of an
individual’s father is m(mf)=m2f
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Warlpiri Tribe
There are 8 sections and each person in a section is
linked to all other sections.
The Warlpiri believe their ancestors set up a pattern
of life.
Sociological structure includes specific relations,
like marriage laws, linking all sections.
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Warlpiri Marriage Rules
Marriage takes place between specific people from
specific sections
Let “=“ indicate marriage:
1=5, 2=6, 3=7, 4=8
A person in section 1 marries a person in section 5
A person in section 2 marries a person in section 6
A person in section 3 marries a person in section 7
A person in section 4 marries a person in section 8
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Warlpiri Kin Relations
Matricycles determine the section of the children
{1,3,2,4} and {5,8,6,7}
Example for {1,3,2,4}
A mother in section 1 will have children in section 4
1→4
A mother in section 4 will have children in section 2
1→4→2
A mother in section 2 will have children in section 3
1→4→2→3
A mother in section 3 will have children in section 1
1→4→2→3→1
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Warlpiri Kin Relations
Continued
Patricycles determine inheritance rights and
responsibilities for land and religious rituals
{1,7}, {2,8}, {3,6}, {4,5}
Example for {1,7}
A man from section 1 must marry a woman from section 5
and their son will be in section 7
1→7
The son must marry a woman from section 3 and their son
will be in section 1
1→7→1
Marriage Laws 1=5, 2=6, 3=7, 4=8 Matricycles: {1,3,2,4} {5,8,6,7}
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Kin Relations and Group
Theory
In general, begin labeling in one section
Section 1 = I
Label all composite relations based on matricycles
and marriage laws
Represent relationships with letter symbols
e.g. m=mother, mf= mother of father
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Procedure
Matricycles {1,3,2,4} and {5,8,6,7}
Marriage Laws 1=5, 2=6, 3=7, 4=8
I, identity, in section 1.
Mothers of I, m, in section 3.
Mothers of mothers of I, mm=m2, in section 2
Mothers of mothers of mothers of I, mmm=m3, in section
4
Fathers of I, f, in section 7.
Mothers of fathers, mf, in section 5.
Mothers of mothers of fathers, mmf=m2f in section 8
Mothers of mothers of mothers of fathers, mmmf= m3f, in
section 6
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Warlpiri Kin Relations Table
I
m
m2
m3
f
mf
m2f m3f
I
I
m
m2
m3
f
mf
m2f
m3f
m
m
m2
m3
I
mf
m2f
m3f
f
m2
m3
m2
m3
m3
I
I
m
m
m2
m2f
m3f
m3f
f
f
mf
mf
m2f
f
f
m3f m2f
mf
I
m3
m2
m
mf
mf
f
m3f
m2f
m
I
m3
m2
m2f m2f
mf
f
m3f
m2
m
I
m3
m3f m3f
m2f mf
f
m3
m2
m
I
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Warlpiri as a Group
Use Table to analyze properties of groups
Closure: every result is one of the original 8 sections
Identity element: All items in column I replicate row labels,
and the row I replicates column headings
Unique inverse: Identity element, I, occurs once in each
row and each column, therefore there exists an inverse
linking each section to identity
• m-1=m3
Associativity: Consecutively applied relationships give the
same result, no matter what order they are applied
• mmf= (mm)f = m(mf) = m2
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Warlpiri as D8 Group
A dihedral group is symmetry group for an n-sided regular polygon for n >
1, denoted Dn. The group order of Dn is 2n.
The structure of the Warlpiri Tribe is analogous to the symmetries of a
square, n=4-agon
Take the standard position of I
m, f are generators:
take m= clockwise rotation through 90º ,
f=a reflection across a horizontal line
All other positions are combinations of m,f
Defining Relations:
m4=I (matricycle)
f2=I (patricycle)
(mf)(mf)=I (marriage rule) spouse of a one’s self is one’s self
Definition taken from MathWorld Website:
http://mathworld.wolfram.com/search/
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Cayley Diagram
Vertexes are group elements
Mother relations are solid edges
Father relations are dotted edges
Generators are edges connecting them
Directed graph: moving against the
arrow represents inverse relations.
Clear to see matricycles and patricycles
Image taken from Ascher, Marcia. Ethnomathematics. (76).
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Notes on Warlpiri as D8
Every composite pairing returns one of 8 sections:
close linkages.
Each element in the group occurs once in every
row: links each section through descent
relationship.
Parents, aunts, uncles, and first cousins do not
belong to section with which child can marry.
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Familial Terminology
We are specifically dealing with relationships
between descendents of common ancestor at
the same generation
First cousin- Children of siblings
Second cousin- Children of first cousins
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M-1 = W-5
M-7 = W-3
M-5 = W-1
M8=W4
M4=W8
M-3 = W-7
M-1 = W-5
M7=W3
M3=W7
M-2 = W-6
M4=W8
M8=W4
M-6 = W-2
M7=W3
M3=W7
Marriage Laws: 1=5, 2=6, 3=7, 4=8
Matricycle: {1,3,2,4} {5,8,6,7}
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Sociological Interest
Warlpiri’s Sociological Structure
Complex social structure carried through
generations
No distant relations because everyone is
interconnected
Naturally deters incestuous relationships
Contrast with U.S.
Less rigid marriage laws afford a diverse social
structure
First cousins can marry in VT and NY
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Contrast with U.S.
Vermont:
§ 1. Man forbidden to marry relatives
A man shall not marry his mother, grandmother, daughter, granddaughter,
sister, brother's daughter, sister's daughter, father's sister or mother's
sister. (Amended 1975, No. 28, § 1.)
§ 2. Woman forbidden to marry relatives
A woman shall not marry her father, grandfather, son, grandson, brother,
brother's son, sister's son, father's brother or mother's brother.
(Amended 1975, No. 28, § 2.)
New York:
§ 255.25 Incest. A person is guilty of incest when he or she marries or
engages in sexual intercourse or deviate sexual intercourse with a person
whom he or she knows to be related to him or her, either legitimately or
out of wedlock, as an ancestor, descendant, brother or sister of either the
whole or the half blood, uncle, aunt, nephew or niece. Incest is a class E
felony.
Information gathered from the Vermont Statutes Online:
http://www.leg.state.vt.us/statutes/fullchapter.cfm?Title=15&Chapter=001
The New York State Sexual Statutes:
http://www.lectlaw.com/files/sex09.htm
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Resources
Ascher, Marcia. Ethnomathematics: a multicultural view of mathematical
ideas. U.S.A.: Brooks/Cole Publishing Company (1991).
Vermont Online Statutes
http://www.leg.state.vt.us/statutes/fullchapter.cfm?Title=15&Chapter=
001
New York State Sexual Statutes
http://www.lectlaw.com/files/sex09.htm
Definition taken from MathWorld Website:
http://mathworld.wolfram.com/search/
Joanna Ellis-Monaghan
James Hefferon
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