#projective
METEORITE
Folk Names: Aerolith, Aerolite
Energy: Projective
Planet: none, meteorites are associated with the Universe
Elements: Akasha, Fire
Deity: The Great Mother
Associated Stones: Peridot, Diamond
Powers: Protection, Astral Projection
Magical/Ritual Lore:
Meteorites have long been held in fascination by humans. They have been thought to be gifts from the gods and goddesses. Certain meteorites, such as the Kaaba stone in Mecca and a stone thought to represent the Great Mother Goddess of Phrygia, have been worshipped as symbols of divinity.
A four-ton stone has been revered in China as a holy object since the 1200’s. The stone, shaped like a crouching ox, resides in a Bhuddist shrine. Recently, however, a team of Chinese geologists studied
the stone and determined it to be a meteorite that landed about 1,300 years ago. The stone is no longer worshiped. In Babylon the meteorite was a powerful magical protectant. It was thought to remove all evils due to its strange appearance and the "roar of its awful might.“
Peridot is often found in meteorites. I held a small cut meteorite recently and studied the green peridot crystals that were packed inside it. The stone was worth about $3,000, so it didn’t go home with me. Recently, tiny diamonds were found inside meteorites that fell in Mexico in 1969-the first discovered that had formed off of our planet.
At one place or another on Earth, meteorites were used to explain the origin of life. If rocks fell to the earth from space, so too could plants, water, animals and people.
Symbolically, meteorites can be viewed as the spiritual penetrating the physical, as astral power, divine order or whim, though a friend of mine says they’re the melted remains of spaceships from distant galaxies!
Magical Uses:
Meteorites are unearthly things, literally. They possess the powers of intergalactic flight, of movement, of speed, and of energy unhindered by gravity.
Use them in rituals of protection. Place one on the altar near white candles; or carry in the hand.
They are also called upon to promote astral projection. A small meteorite or a fragment of one is placed beneath the pillow during attempts at conscious astral projection.
Yes, they are available for sale at reasonable prices. I visited the Reuben H. Fleet Space Theatre’s gift shop in San Diego a few days ago and found small meteorites for $3.00.
[Source: Cunningham’s Encyclopedia of Crystal, Gem, & Metal Magic by Scott Cunningham]
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PYRITE
Folk Names: Fool’s Gold, Pyrites, Iron Pyrite
Energy: Projective
Planet: Mars
Element: Fire
Powers: Money, Divination, Luck
Magical/Ritual Lore:
Pyrite was used by ancient Mexicans in fashioning polished mirrors, which may have been used to divine the future. Pieces of this strange mineral were also placed in American Indian shamans' medicine bundles, perhaps to lend extra energy.
In ancient China this stone was used to guard against crocodile attacks, a problem which, fortunately, most of us seem to avoid without the stone.
Magical Uses:
Popularly known as fool’s gold, pyrite is often found associated with real gold. So who, exactly, is the fool?
Because of the yellowish shimmer and shining nature of this "stone,“ it is used to draw wealth and money. Set five pieces of pyrite on your altar. Surround them with five green candles. Light the candles and visualize money coming your way, fulfilling your monetary needs.
Pyrite is also carried to bring money and luck.
A flat, shimmering surface of pyrite can be used as a magic mirror to awaken psychic impulses. Carried, it is a luck-bringer.
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Making matrices better: Geometry and topology of polar and singular value decomposition
- Dennis DeTurck
- Amora Elsaify
- Herman Gluck
- Benjamin Grossmann
- Joseph Hoisington
- Anusha M.Krishnan
- Jianru Zhang
2017
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You don’t want to view matrices as 9 entries in an array. So you learn the inner product, rank, and the determinant. Then how do matrices look?
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The two components of O(3) appear as real projective 3-spaces in the 8-sphere, each the core of a open neighborhood of nonsingular matrices, whose cross-sectional fibres are triangular 5-dimensional cells lying on great5-spheres. The common boundary of these two neighborhoods is the 7-dimensional algebraic variety V⁷ of singular matrices.
This variety fails to be a submanifold precisely along the 4-manifold M⁴ of matrices of rank 1. The complement V⁷−M⁴, consisting of matrices of rank 2, is a large tubular neighborhood of a core 5-manifold M⁵ consisting of the “best matrices of rank 2”, namely those which are orthogonal on a 2-plane through the origin and zero on its orthogonal complement.
V⁷ is filled by geodesics, each an eighth of a great circle on the 8-sphere, which run between points of M⁵ and M⁴ with no overlap along their interiors. A circle’s worth of these geodesics originate from each point of M⁵, leaving it orthogonally, and a 2-torus’s worth of these geodesics arrive at each point of M⁴, also orthogonally.
