| Tikalon Blog is now in archive mode.
An easily printed and saved version of this article, and a link
to a directory of all articles, can be found below: |
|
This article |
| Directory of all articles |
The Seven Sisters
March 22, 2021
Ancient geometers accomplished a lot by using their simple
tools of the
straightedge and
compass. This included
construction of
regular polygons from few
sides, such as the simple
equilateral triangle, to others with a far greater number of sides, such as the
dodecagon (12 sides) and the
icosagon (20 sides). You can create the icosagon by
edge-
bisection of the
regular decagon (10 sides), and likewise create a 40-sided polygon from an icosagon,
ad infinitum.
Regular polygons, from the equilateral triangle (3 sides) through the regular decagon (10 sides). The colors are based on the resistor color code for the number of sides. All but the seven-sided heptagon can be constructed using a straightedge and compass. (Created using Inkscape. Click for larger image.)
Except for the seven-sided
heptagon, the regular polygons of sides three through ten can be constructed using a straightedge and compass. The eleven-sided
hendecagon is also non-constructible. As first stated by
Carl Friedrich Gauss (1777-1855) and later
proved by
Pierre Wantzel (1814-1848), a
polygon is only constructible if it has a number of sides that's a
product of a
Fermat prime[1] and a
power of two (this allows the bisection mentioned above). Fermat primes are given by
n = 1 + 2(2k),
where k = 0 gives us a triangle, while k = 1 gives us a
pentagon.
Despite its
ill omen in
geometry, the
number seven is considered by many to be a
lucky number. It also has a
preeminent place in human culture, being associated with many, and diverse, items, some of which are listed below:
• The seven deadly sins (lust, gluttony, greed, sloth, wrath, envy, and pride)
• The seven virtues (chastity, temperance, charity, diligence, kindness, patience, and humility
• The seven classical planets (The Sun, Moon, Mars, Mercury, Jupiter, Venus, and Saturn)
• The Pleiades, the seven daughters of Atlas
• The seven metals of antiquity (gold, silver, iron, mercury, tin, copper, and lead)
• The seven heavens
• The seven days of the week
• The seven colors in the rainbow
• The seven seas
• The seven continents
• The seven climatic zones
• The seven wonders of the ancient world
• The seven notes in the Western (diatonic) musical scale
The seven classical planets (The Sun, Moon, Mars, Mercury, Jupiter, Venus, and Saturn) are associated with the seven metals of antiquity. These are, respectively, gold, silver, iron, mercury, tin, copper, and lead. (Modified NASA planetary images, Sun image from SOHO - EIT Consortium, ESA, and NASA, and Lunar Reconnaissance Orbiter image by NASA/GSFC/Arizona State University. Click for larger image.)
Among these groups of seven are the Pleiades, the seven daughters of Atlas and the
Oceanid nymph,
Pleione. The Pleiades, who are also nymphs, are often called the "Seven Sisters," and they are companions of
Artemis, the
Greek goddess of the
hunt and the protectress of
young girls. I wrote about the Pleiades in an
earlier article (The Pleiades, September 24, 2014).
Pleiades is also the name of a conspicuous
star cluster, listed also as
Messier 45 and
NGC 1432/35. The
stars in the Pleiades are quite close to us by
astronomical standards, and these stars are easily
seen with the
unaided eye. In fact, the ability to
resolve the individual stars of the Pleiades was an ancient
eye test. The
Hubble Space Telescope has
measured the distance to the Pleiades as 440
light years.[2]
The Pleiades of Greek Mythology (left) and the Pleiades of Astronomy (right). The mythological Pleiades were Alcyone, Celaeno, Electra, Maia, Merope, Sterope, and Taygete. (Left image, The_Pleiades, an 1885 oil on canvas painting by Elihu Vedder (1836–1923), located at the Metropolitan Museum of Art (Accession number, 10.64.13, a Gift of George A. Hearn, 1910). Right image, via Wikimedia Commons. Click for larger image.)
Ray P. Norris of
Western Sydney University and
CSIRO, and
Barnaby R. M. Norris of the
Sydney Institute for Astronomy and the
University of Sydney, all located in
New South Wales, Australia, have recently posted an article on
arXiv that examines two interesting observations about these seven sisters.[3] One of these is that they are called the "Seven Sisters" despite the fact that most people with good
eyesight can see only six stars. The other is that the mythology of these sisters is similar across vastly separated
cultures, including
Australian Aboriginal cultures that were assuredly isolated from
Greek culture. The Australian authors find that these observations can be jointly explained by the idea that the seven sisters mythology predates the
migration of humans out of Africa around 100,000
BC.[3]
In Greek mythology, Pleione is the mother of the Seven Sisters, but the star,
Pleione is the seventh in brightness and is possibly the missing seventh star of the Pleiades (see figure).[3] Stars that are close to Earth, like those of the Pleiades, exhibit a large
proper motion caused by a combination of their
rotation and the Sun's rotation around the
galactic center. This means that their observed
position in the
night sky will change over long periods of time. Pleione is presently just five
arc-minutes from the star,
Atlas, which is about four times brighter than Pleione, so the resulting
glare from Atlas prevents most people from seeing Pleione. As can be seen in the figure, the separation was slightly greater around 100,000 BC.[3]
The locations of the Pleiades at present, and at 100,000 BC. The area of each circle is proportional to the apparent magnitude of the star. (Fig. 1 of ref. 3 (modified).[3] Click for larger image.)
The Pleiades mythology has many similarities across cultures. There are seven sisters, and these are all
young girls.
Orion is identified as a
hunter, a
young man, or a group of young men, who attempt to catch or
rape these young girls. These elements are also found in the mythology of Australian Aboriginal cultures.[3]
The Pleiades are also an element of Aboriginal
calendars. Their
heliacal rising, the
annual rising when they are briefly visible above the
eastern horizon just before
sunrise after being hidden behind the Sun, marks the start of
winter.[3]
References:
- Sequence A019434, Fermat primes, at The On-Line Encyclopedia of Integer Sequences.
- Hubble Refines Distance to Pleiades Star Cluster, Space Telescope Science Institute (STScI) News Release no. 2004-20, June 1, 2004.
- Ray P. Norris and Barnaby R. M. Norris, "Why are there Seven Sisters?" arXiv, December 18, 2020.
Linked Keywords: Greek mathematics; ancient geometer; tool; straightedge; compass (drawing tool); straightedge and compass construction; regular polygon; edge (geometry); side; equilateral triangle; dodecagon; icosagon; bisection; regular decagon; ad infinitum; electronic color code; resistor color code; heptagon; Inkscape; hendecagon; Carl Friedrich Gauss (1777-1855); conditions for constructibility; prove; Pierre Wantzel (1814-1848); multiplication; product; Fermat prime; power of two; pentagon; ill omen; geometry; number seven; luck; lucky; symbolism of the number 7; human culture; seven deadly sins; lust; gluttony; greed; sloth; wrath; envy; pride; seven virtues; chastity; temperance (virtue); charity (virtue); diligence; kindness; patience; humility; seven classical planets; Sun; Moon; Mars; mercury (planet); Jupiter; Venus; Saturn; The Pleiades; daughter; Atlas (mythology); seven metals of antiquity; gold; silver; iron; mercury (element); tin; copper; lead; seven heavens; seven days of the week; seven colors in the rainbow; seven seas; seven continents; seven climatic zones; seven wonders of the ancient world; seven notes in the Western (diatonic) musical scale; NASA planetary image; SOHO - EIT Consortium, ESA, and NASA; Lunar Reconnaissance Orbiter; NASA/GSFC/Arizona State University; Oceanid; nymph; Pleione (mythology); Artemis; Greek mythology; Greek goddess; hunting; hunt; virginity; young girl; Pleiades star cluster; Messier object; Messier 45; New General Catalogue; NGC 1432/35; star; astronomy; astronomical; visual perception; sight; naked eye; unaided eye; angular resolution; resolve; eye examination; eye test; Hubble Space Telescope; astrometry; measurement; light year; Pleiades; Alcyone; Electra; Maia; Merope; Sterope; Taygete; oil painting; oil on canvas painting; Elihu Vedder (1836–1923); Metropolitan Museum of Art; Wikimedia Commons; Ray P. Norris; Western Sydney University; CSIRO; Barnaby R. M. Norris; Sydney Institute for Astronomy; University of Sydney; New South Wales, Australia; arXiv; eyesight; culture; Aboriginal Australians; Australian Aboriginal; Ancient Greece; Greek; Recent African origin of modern humans; migration of humans out of Africa; Anno Domini; BC; Pleione (star); proper motion; rotation; galactic center; astrometry; position; night sky; arc-minute; Atlas (star); glare (vision); locations of the Pleiades at present, and at 100,000 BC; area; circle; proportionality (mathematics); proportional; apparent magnitude; Orion (constellation); young man; rape; calendar; heliacal rising; year; annual; east; eastern; horizon; sunrise; winter; Sequence A019434; The On-Line Encyclopedia of Integer Sequences.