Circles, Spheres and Balls
In hyperbolic geometry the circumference of a circle of radius r is greater than 2πr. It is in fact equal to
The area of the enclosed disk is
The surface area of a sphere is
The volume of the enclosed ball is
For the measure of an n-1 sphere in n dimensional space the corresponding expression is
where the full n dimensional solid angle is
using for the Gamma function.
The measure of the enclosed n ball is:
Read more about this topic: Hyperbolic Geometry
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And you O my soul where you stand,
Surrounded, detached, in measureless oceans of space,
Ceaselessly musing, venturing, throwing, seeking the spheres to connect them,
Till the bridge you will need be form’d, till the ductile anchor hold,
Till the gossamer thread you fling catch somewhere, O, my soul.”
—Walt Whitman (1819–1892)
“I count those feathered balls of soot
The moor-hen guides upon the stream,
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And turn towards my chamber, caught
In the cold snows of a dream.”
—William Butler Yeats (1865–1939)