Number 9...Number 9...Number 9...
Mathematics: Not only is 9 a square, but anyone who learned their base 10 times table will recall that a number can only be divided by 9 if the sum of the digits reduces to 9 (ya know, like 18 -> 1 + 8 = 9, 27 -> 2 + 7 = 9, etc.). It's also an exponential factorial (3^2^1 = 9). In the base 10 logarithm used for probability, nines are used to express the probability of an event's opposite. Though pi's decimal places are random, the first repeat occurs from positions 762 to 767 -- 9 repeated 6 times -- and the repeat is commonly called the Feynman Point.
Enneagrams: The enneagram is used to diagram nine different personality stereotypes.
Nine Muses: (From http://www.greatdreams.com/nine/nine.htm) "in Greek mythology, the nine patron goddesses of the arts; daughters ofZEUS and Mnemosyne, a TITAN who personified memory. They were: Calliope (epic poetry and eloquence), Euterpe (music and lyric poetry), Erato (love poetry), Polyhymnia (oratory or sacred poetry), Clio (history), Melpomene (tragedy), Thalia (comedy), Terpsichore (choral song and dance), and Urania (astronomy)."
To the highest extent: Many cultures view nine as the highest attainable limit to something, such as "cloud nine" or "dressed to the nines".
Chemistry: On the Periodic Table, Fluorine has Atomic Number 9, so it has nine protons and nine electrons. http://www.lateralscience.co.uk/Fluorine/Fluorine.html calls fuorine the "gas of Lucifer".... The site also gives accounts of how difficult it is to isolate without dying or suffering serious injury. Alchemists often used fluorine as a solvent, since it makes metals flow better. It's a pale green gas that makes me think of the green light that's emmitted when someone casts the Killing Curse.
I'll keep track of the number 9 throughout the HP series....
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