C H A R L E S P. F U N K H O U S E R
Heptades and
Heptagons:
The Historical Roots of
R
OLLING A 7 WITH A PAIR OF DICE,
breaking a mirror and getting cautioned about 7
years of bad luck, drinking 7-Up, flying in a Boeing 747, working at something 24/7, staying
open 7 days a week: What is it about the number 7 that has made it such a common part of our vocabulary and daily lives? Whether discussing good or
bad fortune, measuring time, or exploring a numerical framework within religions of the world, the number 7 has a special place in the world of numbers, numerals, and culture. How 7 took on such a key role in
world cultures and mathematics is a rich and diverse
story spanning all seven continents.
Historical Perspectives
IN BABYLONIA AND THE WHOLE REGION OF
Mesopotamia, 7 was the sign of wholeness and
plenty. The Babylonion pyramidal temple, the ziggurat, had 7 steps or stories to remind the observer
of the “house of the 7 parts of the world” and the “7
heavenly spheres.” The Babylonian Tree of Life had
7 branches each with 7 leaves. It is thought that this
shape was the model for the 7-branched candelabra
used by the Jewish faith (to be discussed later). In
ancient Egyptian lore, there were 7 paths to heaven
and, by doubling 7, one could list the 14 places in
the Realm of the Dead. It was here that Osiris led
his father through the 7 halls of the underworld.
Other early civilizations, such as the Greeks, Romans, and Chinese, counted 7 planets: sun, moon,
Mercury, Venus, Mars, Jupiter, and Saturn—even
though the sun and moon are not planets and Earth
is a planet. Early astronomers saw these “planets” as
being embedded in 7 heavenly spheres later discussed by English Romantic poets. The Greek
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7
mathematician Hippocrates called 7 the number of
cosmic structure and suggested that 7 is “the dispenser of life and is the source of all change,” as
most directly shown by the phases of the moon
changing every 7 days.
It is of interest to note that the Greek word for
week is hebdomada or “seven-ness”; in Latin, it is septimana; in Italian, settimana; and in French, semaine—
each word being related to the number 7. The names
of the days of the week, including those of the Jewish
and early Christian societies, originally were based on
this 7 count. Current day names in English were
taken from German and Romance languages. That is,
Sunday was named for the first “planet,” the sun; Saturday was named for the seventh “planet,” Saturn.
Nevertheless, one still can see a mixture of the counting and planetary origins of the day names in presentday German: Montag (“moonday”) for Monday,
CHARLES FUNKHOUSER, [email protected],
teaches at California State University at Fullerton,
Fullerton, CA 92834. He is interested in appropriate uses
of technology to develop mathematical problem solving,
methods of teaching mathematics to special learners, and
ethnomathematics.
Edited by DAWN ANDERSON, danderson@exchange
.fullerton.edu, Department of Secondary Education,
California State University, Fullerton, 800 North State
College Boulevard, Fullerton, CA 92834
This department explores rich mathematical ideas by revisiting their origins and early investigations found in the history
of mathematics. Authors interested in contributing to this
department should send manuscripts to the editor, c/o NCTM,
1906 Association Drive, Reston, VA 20191-1502.
MATHEMATICS TEACHING IN THE MIDDLE SCHOOL
Copyright © 2003 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.
This material may not be copied or distributed electronically or in any other format without written permission from NCTM.
Mittwoch (“midweek”) for Wednesday, and
Sonnabend (“eve of the sun”) for Saturday.
The Mayans also believed in a 7-layered sky.
They further believed that a woman was represented
by 3, a man by 4, so that together they produced the
unity of life, 7. In a similar, but reversed representation, Asian and European cultures assigned 3 to the
masculine, 4 to the feminine, and their marriage resulted in the unity, 7. The Plains Indians used 7 in
constructing an important religious and astronomical symbol, the medicine wheel. Using their sacred
number 4 (symbolizing the directions of the winds
and change), 7 was multiplied by 4 to generate 28
spokes that radiated from its center to the rim of the
medicine wheel. These 28 spokes are said to represent the 28 ribs of the bison, an animal sacred and
important to Native American cultures. In addition, a
number of these rib spokes are placed along the line
of occurrence of important astronomical events in
the lives of Native Americans, such as the beginning
of seasons or the time for traditional buffalo jumps.
Religious Perspectives
RELIGIOUS REFERENCES TO 7 ARE MANY AND CUT
across many cultures and faiths. The Old Testament
says that the world was created in 7 days. Cain’s
murder of Abel was to be avenged 7 times. Lamech
was born 7 generations after Adam and lived 777
years. Seven steps led to King Solomon’s Temple.
Noah’s dove stayed away for 7 days. Penances, punishments, and wedding blessings are to be repeated
7 times. There were 70 nations and judges of the
Sanhedrin. The 7-branched Jewish menorah is filled
with symbolic meaning; in fact, the numbers assigned to the letters of the 7 branches can be added
and then retranslated using a traditional Jewish algorithm to yield Yahweh, the Hebrew name for God.
Many New Testament references to the number 7
are found, as well. Believers are cautioned that each
should forgive 70 times 7. Christ spoke 7 last words.
The Book of Revelation prophesizes 7 horns of two
kinds: a lamb with 7 horns and 7 horns blowing on the
Day of Judgment. As an expression and interpretation
of the New Testament, the Catholic faith holds to 7
sacraments, 7 gifts of the Holy Spirit, and 7 deadly sins.
Furthermore, the Catholic Mass is divided into 7 parts.
What might the reason be for all these biblical references to 7? In addition to the obvious antecedents from
earlier cultures, German theologians speculated that
the indivisibility or prime nature of 7 was the basic reason: both vengeance and forgiveness should be unchanging and indivisible, and 7 as a prime number was
a useful representation of these characteristics.
The number 7 also finds frequent expression in the
Moslem religion and its central text, the Qur’an. The
Islamic profession of faith consists of 7 words: la ilaha
illa Allah Muhammad rasul Allah (translated often as
“There is no God but Allah, and Muhammad is his
prophet”). The Qur’an states that God created heaven
and earth in 7 layers. During a hajj (a religious pilgrimage during Ramadan), the Kaaba monument in
Mecca is circled 7 times. At a certain station along this
pilgrimage, the pilgrim says 7 times, “Allahu akbar!
(God is greater than everything!)” At the end of the
hajj, the devil is stoned by 3 volleys of 7 stones each.
The extended idea of 7 reaches an extreme with the
Persian and Iraqi Sufis in the image of 70,000 veils of
light and darkness, which are to separate humans
from God. This spiritualist Islamic group believes that
God is praised by beings with 70,000 heads, each of
which has 70,000 faces, 70,000 mouths, 70,000
tongues, and each of which speaks 70,000 languages.
Modern Perspectives
THE ARTS, LITERATURE, AND PHILOSOPHY ALSO
have many references to the number 7. In classical
studies, seven Liberal Arts sections were delineated:
grammar, rhetoric, dialectics, music, arithmetic,
geometry, and astronomy. Seven notes on the musical
scale return to the first one in the octave. In literature,
Shakespeare wrote of the 7 ages of humankind. English Romantic poets, such as Keats and Donne, had
poetic images of the music of the 7 spheres. The English author Sir Thomas Browne suggested that every
seventh year of life brings some important change in
mind or body (7 as the age of reason; 14, the age of puberty; 21, adulthood . . .). Similarly, the Arab philosopher and physician Ibn Tufayl expressed the idea that
human development occurs in stages of 7 years. He
wrote a novel titled Hayy ibn Yaqzan in which the hero
develops moral and spiritual perfection in periods of 7
years. Last, in both Eastern and Western literature,
the expression “seventh son of a seventh son” is used
to designate an especially fortunate individual.
The number 7 continues to play an important role
in contemporary cultures today. For example, the
African American celebration of Kwanzaa uses a 7tiered candelabra as its celebratory centerpiece. This
centerpiece for the December festival holds 3 red candles, 3 green candles, and 1 black candle, each symbolizing a different traditional pan-African value: unity,
self-determination, collective work and responsibility,
cooperative economics, purpose, creativity, and faith.
In popular culture, 7 often plays a special role. Was it
by chance that Ian Fleming named his special agent
007? It was quite possibly to suggest that James Bond
had some special, almost superhuman qualities.
In summary, reflecting back on the roots and history of 7, we see that 7 is frequently invoked in diverse
societies, faiths, and times. One group might speak of
V O L . 9 , N O . 2 . OCTOBER 2003
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7 as a symbol of luck, wholeness, and plenty, whereas
another might use 7 to evoke misfortune and the
number of halls in the underworld. Likewise, 7 has described the Tree of Life and the heavenly spheres, or
it has warned of coming penance and plagues. With
colorful and powerful images like these, it is no wonder that 7 has played such a notable role in our mathematical lexicon.
The next time you see or hear the number 7,
whether in someone’s exclamation of (good or bad)
luck or on a soft drink can, remember the roots of
7, which reach back to the beginnings of history, religion, and culture.
Teacher Notes and Solutions to the
Student Activity Sheet
prealgebra skills in working with decimals.
1. If one takes the number of heads × faces ×
mouths × tongues × languages, one would get
(70,000)5 = 75 × 10,000 5 = 7 5 × 1020, which is
16,807 followed by twenty zeroes.
2. No, because 7 does not divide 365 without a remainder.
3. 365 = (52 × 7) + 1, so each day or holiday shifts
by one day, assuming no leap years. In 2003,
New Year’s Day was on a Wednesday, so in
2004, it will be on a Thursday. Since 2004 is a
leap year, the shift would be two days following
February 29, and 366 = (52 × 7) + 2, so it will be
on a Saturday in 2005.
QUESTIONS 1–4 OF PART A ARE DESIGNED TO
enable students to see connections between number
and geometry, in particular, probability and geometry.
1. This exploratory exercise familiarizes students
with heptagons, so strategies and suggestions
will vary. The use of small groups or pairs is especially helpful with this question.
2. Students might try to induce a pattern or formula
to get this measure from equilateral triangles,
squares, pentagons, and hexagons. The formula induced would be [(n – 2) • 180o]/n, where n is the
number of sides of the polygon. In this case, n = 7,
and the angle degree measure is 128 4/7 degrees.
3. No, because 7 does not divide 360 degrees. Either a hexagon or octagon would work, because
both 6 and 8 divide 360 degrees, a condition for
tessellating a plane.
4a.2/6 = 1/3. Note: Most would think 2/7, but this
problem is like a sitting-in-a-circle or key-ring
probability problem. If a vertex is fixed, 6 vertices are left from which to select; 2 of those
would be adjacent to the fixed vertex to form a
side. Students can label the vertices of a heptagon and list all 21 possible line segments. Only
7 sides are possible, and 7/21 = 1/3.
4b.4/6 = 2/3. Note: One could use similar reasoning as in 4a, or one could simply take 1 – 1/3 =
2/3, since the two conditions are mutually exclusive probabilities and complementary.
Problems 1–5 of part B are based on the history presented above. They provide an opportunity for students to use critical thinking, practice arithmetic computation, apply properties of exponents, and access
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MATHEMATICS TEACHING IN THE MIDDLE SCHOOL
4. 7 + 7 + 7 = 21 sons. This number is the minimum
because to the seventh son there must be at least
seven males. And there also may have been some
sisters who had sons, too. (Note: Readers might
recognize an opportunity for a “teachable moment” in this question referencing the “seventh
son” expression. Why not “the seventh daughter
of a seventh daughter of a seventh daughter?”)
5. 7/9. The most common way to find this fraction
is as follows: Let n = 0.777, so that 10n = 7.777.
Next, we find 10n – n = 7.777 – 0.777, or 9n = 7,
giving us n = 7/9. (This question could be extended to find 0.1, 0.2, 0.3, and so on.)
Bibliography
The Economist. “Christmas Special: The Chronicles of
Chronology.” The Economist Newspaper Limited 363
(December 22, 2001): 102–3.
Funkhouser, Charles P., Duane A. Porter, and James
Hirstein. “Indian Mathematics: An Ethnomathematical Review.” ERIC Clearinghouse for Science, Mathematics and Environmental Education (August 2002).
ED 438 170.
Johnson, Art. “Math Roots: Historical Roots of Our Calendar.” Mathematics Teaching in the Middle School 8
(December 2002): 196–99.
Karenga, Maulana. “Kwanzaa: A Celebration of Family,
Community and Culture.” Los Angeles, Calif.: National Association of Kawaida Organizations, 2002.
www.officialkwanzaawebsite.org.
Katz, Victor J. A History of Mathematics: An Introduction.
New York, N.Y.: HarperCollins, 1993.
Schimmel, Annemarie. The Mystery of Numbers. Oxford,
U.K.: Oxford University Press, 1993.
Menninger, Karl. Number Words and Number Symbols.
New York, N.Y.: Dover Publications, 1992. Student Activity Sheet
NAME _______________________________________
Part A
The number 7 has some interesting connections to geometry and probability. You probably know that a
stop sign is an 8-sided polygon called an octagon, and that the U.S. military headquarters in Washington,
D.C., the Pentagon, is a 5-sided polygon. But did you know that a 7-sided polygon is called a heptagon? Let’s
investigate some properties of a regular heptagon and how its geometry might connect with probability.
1. Try to draw a regular (equal sided) heptagon using a ruler, protractor, and pencil or by using software, such as The Geometer’s Sketchpad. What are some useful strategies that you might use in
drawing your regular heptagon from your experience drawing other regular polygons like the
square or an equilateral triangle?
2. What is the measure of each angle in a regular heptagon? How can you find this measure from your
work with other regular polygons?
3. Molly is fascinated by the history of 7 and wants to use 7 as a basis for tiling or tessellating the area
of her new patio. Can she do it using regular heptagons? If not, what are the closest regular polygons she might use instead to tessellate the patio?
4. Sean randomly chooses two vertices of a heptagon.
a. What is the probability that the two vertices will form a side of the heptagon?
b. What is the probability that the two vertices will form a diagonal of the heptagon?
From the October 2003 issue of
Mathematics
Teaching
in the
Middle School
Student Activity Sheet
NAME _______________________________________
Part B
A number of other challenging and practical questions from other areas of mathematics can be asked
from the history of 7.
1. Think about the description of the Persian and the Iraqi Sufis’ multiheaded beings. How many languages would be spoken by each being?
2. If we assume that every year has 365 days, does using a week made up of 7 days allow us to exactly
measure out a year? Why or why not?
3. From your answer to the previous question, how does New Year’s Day, your birthday, or the 4th of
July “shift” its day of the week from year to year? For example, New Year’s Day was on a Wednesday
in 2003. On what day will it occur in 2004 and why? What would your answer be for New Year’s Day
2005?
4. Consider the statement “the seventh son of a seventh son of a seventh son.” If Hakeem traces his
heritage back and finds that he is that special son, how many sons are in his family history? Is this
the maximum or minimum number of boys in his family lineage? (Be able to explain your answer to
the second question.)
5. Nha thinks that with all this talk of lucky 7s, the decimal 0.7777 . . ., or 0.7, must have some luck associated with it. She wonders if there is a common fraction that can give her that repeating, nonterminating decimal. What is it? Explain your solution.
From the October 2003 issue of
Mathematics
Teaching
in the
Middle School