Hypersphere
A hypersphere is a higher-dimensional analogue of a sphere.A hypersphere of radius r in n-dimensional Euclidean space consists of all points at distance r from a given fixed point (the centre of the hypersphere). This object is an (n-1)-manifold commonly called an (n-1)-sphere. Hence, the special case of an ordinary sphere in three dimensions would be called a "2-sphere".
The "volume" it encloses is
- <math>\pi^{n/2}r^n\over\Gamma(n/2+1)</math>
where Γ is the gamma function. The "surface area" of this hypersphere is
- <math>2\pi^{n/2}r^{n-1}\over\Gamma(n/2)</math>
See also Hypercube 3-sphere
E'en so the tempest of my grief.html">grief passed by.
With the deliberate purpose of my mind.html">mind,
I gave my dearest hope.html">hope of joy to her.
Had chosen added blessings to confer,
That th' immortal treasures were not mine.html">mine.
Better my lonely sorrow, than to know
Better my grief and my strength to control,
Better to go on, loveless, to the end,
With will most resolute I set my aim
And if I failed to climb the dizzy height,
Till man was formed, and placed upon the sod,
All landscape scenes were lacking in my sight,
In that, all lines of symmetry did meet -
Enthusiasm in abundance, thought,
To help me in my efforts to portray
Stamped on some form, or looking from some face.html">face.
Human emotion for my theme, and make
Anger or Sorrow, Pleasure, Pain, Distress;
The grave.html">grave of my departed hope and love.html">love.html">love.
And soars beyond itself, or selfish things.
Some cheated purpose, some great pain or loss,
Before it labours onward to fruition.
But, as the lark from beds of bloom will rise
Still mounting near and nearer to the light,
Impelled alone by love of upward flight.html">flight,
Some sportman's shot, grazing the singer's throat,
May speed its flight toward the realm of day,
So deathless Genius mounts but higher yet,
Can silence it, or hinder. While there's breath
And lift itself to glory, and to God.
The certain growth of th' upreaching oak.
Talent was mine, not Genius; and my mind
Its selfish love and sorrow.
Did I strive
Of emerald beauty, dark as ocean dyes,
Rose from its grave, and stood by me alive.
The glorious beauty of his face would shine.
.
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