Randomness, Luck, Astragali and Dice
In the time before the mathematical idea of randomness was
discovered, people thought that everything that happened was part
of the will of supernatural beings, the gods, who looked down upon
human affairs and decided to 'tip the balance' one way or another
to influence events. Hence, sacrifices were made and rituals
performed to discover the 'will of the gods' or to try to influence
human affairs. This idea still prevails, and many people all over
the world use lucky charms, engage in superstitious practices, use
horoscopes, and still have some kind of belief that there are such
ways of influencing their lives. The gods may be dead, but 'Lady
Luck' still survives.
The astragalus is a small bone, about an inch cube, found in the
heel of hoofed mammals. Astragali have six sides but are not
symmetrical, so there is no way of knowing which way they will
eventually come to rest. For many ancient civilizations, astragali
were used by priests to discover the opinions of their gods. It was
customary in divination rites to roll, or cast, five astragali.
Typically, each possible configuration was associated with the name
of a god and carried with it the sought-after advice.
Astragali from the heel of a sheep
showing the four positions of rest.
The small one in the foreground is
made from pottery
Showing the four positions of rest. The small one in the
foreground is made from pottery. Astragali found in excavations
typically have their sides numbered or engraved. They were also
used in board games in the First Dynasty in Egypt, c 3500 BCE;
archaeological evidence consists of boards, counters, and astragali
for various games, including one similar to Snakes and Ladders,
still popular today.
The game of Hounds and Jackals
dating from 1800 BCE found in an Egyptian tomb
The astragali have been used from classical times for gambling, and
similar stones are still in use today for games like 'fivestones'
or 'jacks'.
Die A
Gradually, over thousands of
years, astragali were replace by dice [see note 1 below], and
pottery dice have been found in Egyptian tombs. The earliest die
known was made from pottery and excavated in Northern Iraq dating
from about 3,000 BCE. It has dots arranged as in (Die A).
Die B
Die (B), from about 1400 BCE
found in a tomb in Egypt, shows consecutive numbers opposite each
other.
Die C
Dice with other markings like
the names or portraits of gods have been found, probably used for
special games or rituals, and others where some numbers are
repeated, or 'loaded', for special purposes or possibly for
cheating (Die C).
Once the Greeks had worked out the geometry of the polyhedra, dice
of other shapes began to be constructed. However, whether cube or
polyhedral, the shapes were not entirely regular and were therefore
biased.
However, over time, gamblers would get used to using their own
personal dice, and have an intuitive idea of how they would fall,
but given a different set of dice, the odds would be different.
Later, as the manufacture of dice became more exact, and 'true'
dice became more common, some ideas of the frequency of
combinations of numbers began to emerge.
Over time, gamblers would get used to using the same dice, and have
an intuitive idea of how they would fall, but given another set of
dice, the odds would be different. Later, as the manufacture of
dice became more exact, some ideas of the possible combinations of
number began to emerge.
The Earth and The Cosmos
There were many other forms of rituals hoping to overcome the
randomness of nature and man's condition. A few of these which
became of particular mathematical interest are geomancy, the nine
square grid or magic square, and temple designs, the ancestors of
board games.
Geomancy
Geomancy means divination
of or by the earth
, and is a system of 16 mathematically related arrangements of
stones, beans or other available small objects used to make
decisions, answer questions, or foretell the future. The stones are
cast upon the ground and the pattern formed is interpreted. The
symbols represent a series of binary 'opposites' like good and
evil, male or female, sadness and happiness, etc. Combinations of
these opposites can be used to represent odd and even
numbers.
The sixteen figures of the Geomancy
system of Divination. The headings of the columns are: "The greater
fortune" and "The lesser fortune". From a Book of Occult Philosophy
published 1655. Notice that each pair of shapes are associated with
the traditional signs for the planets and that each configuration
could be interpreted from the throws of two dice.
As in all methods of divination, each of these figures has a
number of interpretations depending on its relation to other
figures shown, and many other circumstances like the time of day,
the weather, and the kind of person who is asking the
question.
The Grid of Nine Squares
The Nine square grid is said to come from an ancient system for the
division of land, probably from feudal India. In China the
nine-square configuration was supposed to be an ideal arrangement,
with eight farmers' fields surrounding a central well. The grid of
nine squares, or a circle divided into nine sections by straight
lines often appears as a central form in Tibetan sacred diagrams.
In Scotland, the pattern was used at Beltane (the eve of May) where
eight squares were cut out from the turf, and a bonfire lit on the
central square.
In this way, from practical beginnings in different cultures, the
nine-square grid acquired mystic importance and symbolised divine
order, and the representation of control by the gods.
Magic Squares are directly
related to the Sacred Grid, supposedly being the numerical mystery
which underlies their physical form. The simplest magic square is
the square of nine, ascribed to Saturn, where each row and column
adds up to 15; the total of the rows and the columns is 45, and the
diagonals 30. The 4x4 square with row and column numbers 34 is
assigned to Jupiter, the 5x5 with row or column numbers 65 to Mars,
and so on for the Sun, Venus, Mercury and the 9x9 square with row
or column numbers 369, to the Moon [an NRICH article on Magic
Squares can be found here
].
As with other devices, these magic squares are all said to have
correspondences to different numbers, various deities, days of the
week, natural objects, different qualities, and so on. In the Hindu
Temple Yantra [see note 2 below] you can see the nine squares, the
'sacred space', or source of energy, in the centre.
This is a Yantra from a Hindu
Temple. Yantras (or Mandalas) are used as a focus for mystical
contemplation and often for the basis of design for a temple. This
one is based on a 5 x 5 square, with the 'sacred space' of the 3 x
3 square in the centre.
Board Games are clearly
linked with divination, astrology and sacred geometry, and the
designs of the boards can show their sacred or occult origins. The
popular game of 'snakes and ladders' is controlled by the throw of
dice, and the ladders and snakes originally referring to good and
bad fortune, now refer to good and bad 'luck' in the progress of
the game. In some cases the designs of the boards are the same as
the plans of temples and holy cities with a 'sacred space' in the
centre.
This is the board
for the ancient Korean game of 'Yut' or 'Nyout' The board can be
made of cloth or paper, or can be drawn on the floor. It is played
with four 'Yut Sticks' of semi-circular section, and the way they
fall determines the move of a token. The shape can be square or
circular and represents the division of the world into twenty outer
regions and nine central spaces.
The game of 'Nine
Men's Morris' is played with counters on the dots on this board.
The design is said to represent the four elements, (earth, air,
fire and water) the four winds, or the four cardinal points of the
compass, and the central sacred area was a symbol of rebirth or
renewal.
The game was supposed to have originated in Egypt, and was known to the Romans. This is a picture from the 13th century of the game being played in England.
This is a
traditional diagram used for the Horoscope of Robert Burton from
his tomb in Christchurch in Oxford. This clearly has a link with
the diagrams from sacred architecture and board games.
Mathematics and Magic
In ancient times, few people could understand even the simplest
arithmetic and geometry, and the confusion of mathematics with
magic has a long history.
People who had knowledge of the regular movements of the heavens
were able to predict the position of planets, and the particular
the times when astronomical events appeared in certain sections of
the sky. In ancient civilisations these were highly skilled
technicians, called 'priests', and their activities were partly
scientific, and partly religious. In Europe, after the arrival of
Christianity, the religious aspect of these practices was condemned
as superstition. Because numbers were used in these processes,
anyone who used numbers was regarded with considerable suspicion.
In this way genuine mathematicians were looked upon with suspicion
by the ignorant, and the titles of Astrologer, Mathematician and
Conjurer were virtually synonymous.
An early Bishop of the Church, St. Augustine of Hippo (354-430 CE)
once said:
"The good Christian should beware
of mathematicians and all those who make empty prophecies. The
danger already exists that mathematicians have made a covenant with
the devil to darken the spirit and confine man in the bonds of
Hell."
Augustine was arguing that belief in astrology denies the freedom
of the will.
Roger Bacon (1214 - 1292), often called England's first Scientist,
had a reputation as a 'great necromancer' because of his ingenious
experiments and John Dee (1527 - 1609) probably one of the foremost
mathematicians in Europe of his time, gained a reputation as a
'Conjuror' while he was at Oxford because he was respnsible for
developing a simple mechanical device by which an actor appeared to
fly, and people claimed he was in league with the devil. [see note
3 below]
During the sixteenth century in England, mathematicians like Robert
Recorde (1510-1558) and Thomas Digges (1546-1595) published many
works showing the everyday practical usefulness of mathematical
knowledge for ordinary people clearly showing that mathematics was
not an occult practice.
Roger Bacon
John Dee
Robert Recorde
Following the foundation of the Oxford chairs in mathematics and
astronomy in 1619, some parents kept their sons away from the
university in fear of them becoming contaminated by the 'Black
Art'.
As the predictive power of astronomy and other practical uses of
mathematics became apparent, mathematicians were able to dispel the
idea that many events were not controlled by the goddess Fortuna,
but could be explained in a rational way.
This is the title page of Robert Recorde's The Castle of
Knowledge published in 1556. It is his fourth book for the
self-education of craftsmen and artisans and shows how to make the
instruments for astronomy and navigation. The blind goddess Fortuna
stands on the unstable sphere holding the wheel of chance, while
the Spirit of Knowledge stands on a stable cube holding the
navigator's dividers and the sphere of destiny.
The Beginnings of Probability
Since dice were used in gambling, in religious ceremonies and for
divination, it is believed that those who used the dice had a good
intuitive idea of the likely frequency of various number
combinations. The first printed document showing the possibilities
with three dice was the Latin poem De Vetula , which shows all the
combinations for the fall of three dice, and is believed to have
been written in the early 13th century. The idea of using binomial
coefficients to calculate the possibilities appears in the poem,
but is not taken up until much later [see note 4 below].
Part of the poem De Vetula written in the 13th century,
shows the different combinations of three dice. Notice the written
numerals on the left of the text.The written numerals for the
number combinations appear to the left of the poem.
Girolamo Cardano (1501- 1576)
Since the Christian Church
was against gaming, and there was much superstition about
divination, it is not surprising that a theory of probability did
not begin to appear until the 16th century. Cardano, writing with
considerable personal knowledge of gambling, recognised that if the
die was honest, each face would have an equal chance of appearing.
His manuscript, Liber De Ludo
Aleae , was written about 1526 but only found after his
death, and not published until 1663. He gave tables of the results
for one, two and three dice, but these are not all correct.
However, Cardano is credited with recognising that the abstraction
of the 'honest die' is the key to a theory of probability based on
mathematical principles.
Nicolo Tartaglia (1500-1557)
Tartaglia (1500 - 1557) and others discuss various versions of the
division of the stakes when a gambling game is stopped, called the
'problem of points', and this shows that Cardano's ideas were
likely to be common knowledge among scholars of the later 16th and
early 17th century.
Galileo (1564 - 1642) wrote on probability but his work was not
published until1718. He stated that with three dice there can only
be one way of obtaining a 3 (1,1,1) and an 18 (6,6,6) but there are
three ways of obtaining a 6, and four ways for a 7. However,
although 9 and 12 could be made up in the same number of ways as 10
and 11, from their experience, gamblers agreed that the occurrence
of 10 and 11 were more likely! Galileo showed that the total number
of possible throws with three dice are 216, and he gave a table of
the number of possible throws for a total of 10, 9, 8, 7, 6, 5, 4
and 3, showing that the throws for 11 to 18 were symmetrical with
these. In this way he showed that there were 27 possible throws to
obtain a 10, and 25 for a 9.
This is a copy of the table Galileo
published in his Sopra le
Scoperte de I Dadi (Concerning an Investigation on Dice).
Here he shows clearly how to count the different combinations of
the various throws.
His work showed that by this time
there was no doubt about the general method for calculating chances
with a die, and it was clear that the mathematical concepts of the
equal probability of the throw of a die, and the procedures to
analyse the results were well known.
Pascal's Triangle
By the mid 16th century the theory of probability became
established on a rigorous basis with the work of Pascal and Fermat.
However, as we have seen, the idea of the application of 'Pascal's
Triangle' had been suggested as early as the 13th century but
forgotten for some 200 years. The triangle itself was known and
published before, by Stifel (Arithmetica Integra 1543) Tartaglia
(Trattato 1556) Stevin (Arithmetic 1625) Pierre Herigone (Cours
Mathematique 1634), and we also know it was known to the Chinese
and the Arabs by the mid 13th century, but Pascal was the first to
apply it to probability.
Notes
- The word 'die' (plural 'dice') come from the Latin verb 'dare'
(pron. da-ray) to give its participle dadus means 'given by the
gods'.
- The word Yantra is a Sanscrit word meaning a mystical diagram
or picture. They contain geometric items and archetypal shapes and
patterns of squares, triangles, circles and other floral patterns.
In contrast, a Mantra is a spoken verse or poem.
- At this time, 'Mathematics' included the applied mathematics of
physics, statics, mechanics, hydraulics, and other practical arts.
This is clear from John Dee's famous 'Preface' to the 1570 English
Edition of Euclid.
- It is possible that the scholar who wrote this poem might have
been aware of the 'number triangle' of the Arabs.
References
General Background and History
David, F. N. Games, Gods and Gambling. New York. Dover Books
This is the well-known classic book on the subject still full of interesting and reliable information.
De Moivre, A. (1967) The Doctrine of Chances or A method of Calculating the Probabilities of Events in Play. London. Frank Cass.
A facsimilie of the 1738 second edition of Abraham de Moivre’s classic where he makes corrections, expands the explanations, and gives more details of the solutions of the problems than in the first edition of 1718.
Ore, O. (1953) Cardano, The Gambling Scholar.
A good story and biography of Girolamo Cardano by the Danish Mathematician Oystein Ore. Contains a translation of Cardan’s Liber de Ludo Aleae (Book on Games of Chance).
Hacking, I. (1975) The Emergence of Probability. C.U.P. Cambridge UK
Hacking's book contains many references to original works in the period up to and including the 18th century.
Hald, A. (2003) A History of Probability and Statistics and their Applications Before 1750. New York. Wiley
Pennick, N. (1988) Games of the Gods. London. Hutchinson
Religious Beliefs, Superstition, Astrology and Divination in may cultures and their connections to many games of dice and on the board.
Some Books for the Classroom
Jenkins, G.W. & Slack J.L. (1979) Classroom Experiments with Dice.
St Albans. Tarquin Publications
Woods, G. Symmetry Dice. St Albans. Tarquin Publications
Benson, S. (2005) Ways to think about Mathematics:Activities and Investigations for Grade 6. California. Corwin Press
This is a useful book with many examples of activities. There are sections about probability and binomial coefficients.
Weblinks
Here is a shop for all kinds of dice: large; small; all colours; with numbers; with spots; blank; arithmetic symbols; money symbols; polyhedral; round (yes round!); loaded; and for cheating!
http://www.dice.co.uk/index.htm
This interesting Board Game site has many traditional games like Nine Men’s Morris, various types of strategy games like Solitaire, Fox and Geese; Mancala, Ludo; and Snakes and Ladders. All of them have well-researched historical notes.
http://www.tradgames.org.uk/features/board-games.htm
This site shows a Dice Table for throws of Two Dice and the ways you can calculate the odds for any number combination.
http://www46.pair.com/sengoku/DiceTbl/English/Dice_table.html
Albrecht Durer’s famous engraving of a 4 x 4 Magic Square is explored on this website:
http://mathforum.org/alejandre/magic.square/al.html