Everything is Relative. (Trippy) Consequences of Special Relativity. Derive E = MC2

E = mc². Energy equals mass times the speed of light squared.

We’ve all heard of this equation, but what does it mean? Mass-energy equivalence basically means energy and matter are interchangeable; matter can be converted into energy, and energy can be converted into matter. It’s a consequence of special relativity, a subset of Einstein’s theory of relativity. We see and utilize this equation in everyday life. For example, radioactivity is the emission of energetic particles from unstable atomic nuclei. Mass (atomic nuclei) is being transformed into energy (radiation). Similarly, in nuclear fission (the splitting of atomic nuclei) energy is released through particles and electromagnetic radiation. In nuclear fusion (the fusing of atomic nuclei), the missing mass of the fused nuclei is converted to photons.

Kerning and Hyperbolic Paraboloids

A friend sent me a link to an online kerning game, and I think I might have found my true calling, lulz. I guess my sphere of OCDity extends to include the spacing of letters, even though I’m not a visual person. Arranging and organizing images I can see before me = system running, conceptualizing or manipulating visualizations = system overload. Like I’m truly in awe of people who are good with maps, people who know which direction they’re traveling when they make turns without having to pause and think and confuse themselves in the process.

To segue into math, hyperbolic paraboloids are pretty trippy quadratic surfaces. I know what a hyperbolic paraboloid looks like and I can graph one,¹ but mentally rotating it (interchanging x, y, and z variables), visualizing traces in various coordinates planes, changing the coefficients, shifting the vertex or attempting to extend 2-D conics in 3-D space, results in the aforementioned system overload crashes.

Hyperbolic Paraboloid Rotation

Hyperbolic Paraboloid Flipping: c > 0, c < 0

Critical Point: Saddle Point (Neither Relative Maxima Nor Relative Minima)

Vertical traces (cross sections) are parabolas; horizontal traces are hyperbolas.

Did You Know?

Saddles and pringles are hyperbolic paraboloids.

Notes

¹ Well, admittedly by lazily renaming the x, y, z-axes, yes.

Sources

• http://type.method.ac/
• http://www.math.umn.edu/~rogness/quadrics/hypparab.shtml
• http://www2.math.ou.edu/~pgoodey/classes/honscalc4/notebooks/hypmov.html
• http://academic.brcc.edu/ryanl/modules/multivariable/surfaces/surfaces.htm