Blood Relationship

I considered the relationship between the number of ancestors and blood relationship.

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Japanese Edition
My Ancestors and
Blood Relationship

The number of my ancestors is less than the number calculated by the model 'the nth power of two.'

On my homepage, "My Ancestors, Myself, and My Descendants", I wrote that my ancestors of n-generations ago were 2ⁿ. My son e-mailed me, "The ancestors calculated by 'the nth power of two' model are too much." "The ancestors of the 27th generation ago exceed the present Japanese population." "You should consider blood relationship."

My parents are certainly two persons. But the grandparents are not always four or great-grandparents eight. When there is blood relationship, that is, brother-sister relationship, the ancestors are less than ‘the nth power of two’ model. I will show how the brother-sister relationship reduces the ancestors.

The absence of blood relationship

The number of ancestors

When the marriages are repeated without blood relationship, the direct ancestors are shown as table 1-1.

Table 1-1 The case of no blood relationship

Generation n (ago)

Normal Ancestors

Ancestors (*: one person)

5

32

**

**

**

**

**

**

**

**

**

**

**

**

**

**

**

**

4

16

*

*

*

*

*

*

*

*

*

*

*

*

*

*

*

*

3

8

*

*

*

*

*

*

*

*

Grandparents 2

4

*

*

*

*

Parents 1

2

*

*

Myself 0

1

*

Table 1-1 is summarized to A=2ⁿ (persons).

where, A is the ancestors’ number at n-generation ago

The numbers of the same blood relationship

Under the following assumption, the members of the same blood relationship increase like table 1-2.

• Five genelations ago, there was a couple that has no blood relationship.
• They had two (or three) children.
• After that, their every descendants married with the partner who had no blood relationship.
• Every descendants had two (or three) children.

 

Table 1-2 The members of the same blood relationship

Generation (past)	2 children's case	3 children's case
Starting point 5	One couple
4	2	3
3	2*2=4	3*3=9
2	2*2*2=8	3*3*3=27
Parents 1	2*2*2*2=16	3*3*3*3=81
Myself 0	2*2*2*2*2=32	3*3*3*3*3=243

Table 1-2 is summarized to Bk=kⁿ

where
n: generation(starting point)
k: the number of each couple's children
Bk: The members of the same blood relationship

The Bk exceeds 100 million when the calculation starts from n=27 and k=2, or n=17 and k=3. It means that the Bk exceeds 100 million during the period between 675 and 425 years when one generation is assumed as 25 years.

The presence of blood relationship

The number of ancestors

When blood relationship exists between a husband and his wife, the number of ancestors is shown as table 1-3.

Table 1-3 The brother-sister couple

Generation (past)

Normal Ancestors

Ancestors Including Brother-Sister

Ancestors (*: one person)

5

32

16

**

**

**

**

**

**

**

**

**

**

**

**

**

**

**

**

4

16

8

*

*

*

*

*

*

*

*

*

*

*

*

*

*

*

*

3

8

4

*

*

*

*

*

*

*

*

Grandparents 2

4

2

*

*

*

*

Parents 1

2

2

*

Brother-Sister

*

Myself 0

1

1

*

When my parents have the relationship of brother-sister, their parents, that is, my grandparents are not four but two. So the * marks on the table 1-3, which were calculated by 'the nth power to two' model, do not exist and only * marks exist. The ancestors of the nth generation ago, A_n, are calculated by the following formula;

A_n = 2ⁿ-2^(n-1) = 2^(n-1)

For example: n=5, A_n=2⁵-2^(5-1)=16(persons)

One couple of brother and sister reduces the ancestors to half. It seems to be extremely large effect. But at the same time, I can say that the increasing delays only one generation because the ancestors usually double in one generation.

When my parents have the relationship of cousins, my ancestors are shown as table 1-4.

Table 1-4 The cousins couple

Generation (past)

Normal Ancestors

Ancestors Including Cousin

Ancestors (*: one person)

5

32

24

**

**

**

**

**

**

**

**

**

**

**

**

**

**

**

**

4

16

12

*

*

*

*

*

*

*

*

*

*

*

*

*

*

*

*

3

8

6

*

*

*

*

*

*

*

*

Grandparents 2

4

4

*

*

Brother-Sister

*

*

Parents 1

2

2

*

Cousins

*

Myself 0

1

1

*

On the table, Two of my four grandparents have the brother-sister relationship. So the * marks on the table, which were calculated by 'the nth power to two' model, do not exist.

The ancestors of n generation ago are calculated by "2ⁿ-2^(n-2) = 3?2^(n-2)".
For example: As n=5, ancestors are 3?2^(5-2) = 24.

Table 1-3 and 1-4 are summarized as

Ab=2ⁿ-c*2^(n-g)

where Ab: the ancestors at the n-generation ago.
c: a half of brothers and sisters at the g-generation ago.
n>g.
c<=2(g-1).

The members of the same blood relationship

When the only one couple of husband and wife started to live in an isolated area, extremely close kinship marriages had to be repeated to leave behind their descendants. When the marriage is repeated under the absence of blood relationship, the first couple's descendants increase quickly as shown in table 1-2. This table shows that it will become difficult, sooner or later, to find the marriage partner who has no blood relationship. It is n=4 to 7 (from 100 to 150 years) ago at maximum that we can tell whether the partner has blood relationship or not. The increase of the marriage between the same blood relationship members decreases the increasing speed of the members. But it thickens the blood relationship between the ancestors and their descendants. Besides the blood relationship, there are some cases, such as the birth rate decrease and Natural disasters, which decrease the speed. The decrease of the members of blood relationship is guessed by upsetting table 1-3 or 1-4. Perhaps the members of the same blood relationship extremely decreased many times but recent Japanese population has increased for relatively long time. So I suppose many neighbors, who have thin blood relation, are surrounding me.

A recent newspaper said that according to the DNA analysis, we all are the descendants of a woman who lived in Africa about 200,000 years ago. Without the analysis of my blood relatives' DNA and mine, I cannot easily believe that I am a descendant of an African woman. But table 1-2 shows the possibility of the truth of the news.

I guess that the group she belonged has very small population. And there were no women except her in the small group at that time, or only her group survived and all other groups was extinct. I do not know whether it was the beginning of mankind, or it was the extinction crisis of the mankind. Anyway it is interesting to think that there was an era of very little population.

Rough conclusion

Because there are many people who have various blood relationships, the ancestors of n-generation ago are pretty less than the ancestors calculated by 2ⁿ model. I think we should image as follows:

• The net of the inverted triangle shape spreads exponentially (cf. table 1-1.)

• The net has a lot of tears and rips because of blood relationship (cf. table 1-3,1-4.)

• Many knots (***) on the net are lost.

When the disappeared knots were a great many, I can suppose that there were the periods that two or a few people connect two generations. I am a descendant of the ancestors who were able to be a knot each every time.

If I would use the data about past population changes and take the existence probability of brother-sister relationship into account, I could make a more precise model. But I am satisfied with the knowledge obtained from the four tables. I would like to wait the results of the DNA analysis.

My thought that my ancestors are nearly equal to infinity can be allowed when I think about the beginning of mankind (from 100,000 to 200,000 years ago), and the beginning of the living things (4,000,000,000 or more years ago).

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Uploaded September 4, 2003
Revised December 8, 2003
Revised July 16, 2005