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[Note: If you haven't yet read part one, it is recommended to read it first, before reading this part. Part one is the most recent post here, before this one.]
In part one, a special circumsinusoidal region was described, bounded by parts of semicircles, and a sine or cosine wave. As previously described, this requires that the wave’s amplitude be exactly one-fourth the wavelength. Of course, most sinusoidal waves do not have amplitudes and wavelengths that fall nicely into a 1:4 ratio. What happens, then, with the majority of waves — the ones with with other amplitude:wavelength ratios? Can they still be used for forming circumsinusoidal regions? The answer is yes — but the cost required is that semicircles may no longer by used in their construction.
A semicircle can be thought of as a 180 degree arc, if the diameter, often considered part of its perimeter, is ignored. When the sine or cosine wave’s amplitude:wavelength ratio is not equal to 1:4, the only necessary adjustments for the circular arcs needed to define particular circumsinusoidal regions are identical changes in the angular size of each arc used, plus translations of the centers of the circles which contain each of these circular arcs. These translations move those circle-centers away from the line representing the wave’s rest position, in a direction perpendicular to that line. There are two cases to consider: shorter-amplitude waves (those with an amplitude:wavelength ratio smaller than 1:4), and taller-amplitude waves (where the same ratio is greater than 1:4). The first picture below shows the shorter-amplitude case.
For every half-wavelength which starts at the rest position, an arc through three points is needed for the outer bound of the circumsinusoidal regions, which are shown in yellow. Those three points are two consecutive points (such as A and C above) where the sinusoidal wave crosses the rest position, plus the point at the top of the wave crest (such as B), or the point at the bottom of the trough, exactly half-way along the sinusoidal curve, in-between the two consecutive rest-position points under examination. In this example using points A, B, and C, above, the circle containing those points was constructed by drawing segments AB and BC, and then constructing the perpendicular bisectors of those two segments. Thone perpendicular disectors intersect at some point D, which is the center of a circle containing A, B, and C. The part of this circle which is not used for the arc through A, B, and C is shown as a dashed arc, while the arc used is shown as a solid curve. For other half-waves, to the left or right of this circle and arc, the construction would proceed in the same fashion, but is not shown, for the sake of clarity.
The taller-wavelength case also can be constructed, using the same procedure, as shown below.
These regions, of course, have area. To determine the exact area of any circumsinusoidal region requires integral calculus, and this area is equal to the difference in the areas under two different types of curve. Without calculus, the best that can be found for these areas are mere approximations, not exact answers. I am leaving this find-the-area problem for mathematicians who have a better understanding of calculus than I possess.
The inner boundary of the yellow regions above is a sine curve (technically, a cosine curve, but that’s the same thing, just with a phase shift). The outer boundaries are semicircles. In order for this to work, to form these yellow regions, the semicircle centers (centers of the circles they are each half of) must be directly below peaks, and above troughs, of the sine (or cosine) curve, and vertically positioned at what would be called the rest position in physics. (I’m resorting to use of some physics terminology here, simply because I don’t know the corresponding mathematical terms).
In addition, each semicircle involved must have a radius equal to one-fourth the wavelength of the sine or cosine wave. The two sets of curves cross each other at the rest position, and are tangent to each other at each peak and trough, producing four of these yellow regions per wavelength.
In this case, semicircles could used because I adjusted the wavelength, making it exactly four times the amplitude of the wave. My goal was to compare the two curves, simply to see how well one simulates the other (answer: not very well at all). Then, however, I became more interested in the discrepancy between the two, represented by the yellow regions which are outside the true wave, and inside the semicircles which contain that wave. Until and unless I find that such regions already have a different name, I am naming these two-dimensional curved shapes “circumsinusoidal regions.” There are four of them per wavelength of the wave, and two per semicircle. Each circumsinusoidal region has two vertices, but the two paths connecting them are distinct curves. No part of either path contains any length which is a straight segment.
It would be possible to generate interesting solids by rotating circumsinusoidal regions around vertical or horizontal lines, such as the x- or y-axes, or around diagonal lines. Many such solids would be variations of a torus, including the central hole of a torus, but with circumsinusoidal cross-sections replacing a torus’s circular cross-sections. Unfortunately, I do not have the software I would need to generate pictures of such solid figures.
If the wavelength used for a given sinusoidal wave is not exactly four times the wave’s amplitude, semicircles won’t work to enclose the wave with the same points of tangency, but it is still possible to generate circumsinusoidal regions — using something, in their place, other than semicircles. This will be described in part two, which will be the next post on this blog.
Icosidodecahedral Polyhedron with Irregular Octagons, Rhombi, and Rectangles, as well as Equilateral Triangles and Regular Pentagons, As Faces
This was created using Stella 4d, available at www.software3d.com/Stella.php.
This was created using Stella 4d, available at www.software3d.com/Stella.php.
Please read these safety guidelines carefully. Also, we recommend displaying them prominently, securely fastened to the sturdiest wall in your home, in the event that your gravitational service is ever shut off for non-payment of your GravCorp gravity bill.
Because your friends at GravCorp care about you and your family’s safety, GravCorp will never shut your gravity off abrupty, but does so gradually, over the 24-hour period following the end of the shut-off date (prominently printed in red, bold type) on your gravity shut-off notice. It is best to evacuate early during this period. [Tip: when you notice that you weigh noticeably less than you did the day before, that is your signal to leave.] We are not responsible for anything that happens if you fail to heed this advice, but we do have some safety guidelines to help those who, through no fault of ours, fail to leave their homes in a timely manner.
Once gravity shut-off is complete, if you are still inside your home, follow these safety rules carefully:
1. Be certain to keep moving at all times. Stationary humans have been known to die from lack of oxygen in the absence of gravity, due to the buildup of a spherical cloud of exhaled carbon dioxide, centered in the region of their mouths and noses. If you still have electrical service while your gravity is shut off, however, you can also avoid this danger by turning on all the electric fans in your home, such as the ceiling fan in the picture above.
2. Should you choose to go outside, exercise extreme caution to avoid serious accidents (most of which are likely to be fatal). If you still have telephone or Internet service, we recommend paying your past due GravCorp account balance (plus the $135 reconnect fee) by phone or Internet, from inside your home.
3. Keep all liquids inside containers, for inhalation of even part of a floating ball of water, or other liquid, can cause death by drowning. [Tip: don't forget to seal all toilets -- both bowl and tank -- using approved, waterproof sealing methods and materials.]
4. Act quickly to pay your past due bill, plus the $135 reconnect fee, or have a pressure suit on and pressurized, for the air above you is already beginning to escape into space.
5. Remain calm, do not panic, and consider setting up automatic bank drafts to pay your gravity bill, effortlessly, each month. It’s convenient, safe, and saves you money on postage. (An annual $3.14 convenience fee will be charged to your GravCorp account, on or near July 1st each year, for this optional service.)
[Image credit: The picture above was found at http://www.thedistractionnetwork.com/going-to-bed/.]