### 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:

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.

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