We are examining the spherical coordinate plot of Y equals square root of three over four pi times cosine theta. This function is a spherical harmonic Y one zero. The plot creates a three-dimensional surface where the radial distance from the origin equals the magnitude of the function value. This results in two spheres tangent at the origin along the z-axis.
Let us analyze the mathematical properties of this function. The constant C equals square root of three over four pi, which is approximately zero point four eight eight. For polar angles from zero to pi over two, cosine theta is non-negative, creating the upper sphere. For angles from pi over two to pi, cosine theta is negative, but we take the absolute value, creating the lower sphere.
Now let us observe the geometric construction process. We start with the coordinate axes, then construct the upper sphere for polar angles from zero to pi over two, followed by the lower sphere for angles from pi over two to pi. The key feature is that both spheres are tangent to each other at the origin, creating a distinctive dumbbell-like shape.
This spherical harmonic has important physical interpretations and applications. It appears in quantum mechanics as part of electron orbital descriptions, in electromagnetic field theory, acoustics, and computer graphics. The shape is axially symmetric, meaning it's independent of the azimuthal angle phi, and it's orthogonal to other spherical harmonics, making it useful in mathematical expansions.
To summarize what we have learned: The function Y equals square root of three over four pi times cosine theta creates a distinctive 3D surface consisting of two spheres tangent at the origin. This represents an important spherical harmonic with applications in quantum mechanics, electromagnetics, and wave theory, demonstrating the mathematical beauty of 3D coordinate transformations.