# A Graph Showing Approximate Mass-Boundaries Between Planets, Brown Dwarfs, and Red Dwarf Stars I found the data for this graph from a variety of Internet sources, and it is based on a mixture of observational data, as well as theoretical work, produced by astronomers and astrophysicists. The mass-cutoff boundaries I used are approximate, and likely to be somewhat “fuzzy” as well, for other factors, such as chemical composition, age, and temperature (not mass alone), also play a role in the determination of category for individual objects in space.

Also, the mass range for red dwarf stars goes much higher than the top of this graph, as implied by the thick black arrows at the top of the chart. The most massive red dwarfs have approximately 50% of the mass of the Sun, or about 520 Jovian masses.

# A Graph of Infections and Deaths During the First Four Months of the 2014 Ebola Outbreak Source of data for this graph:  http://www.abc.net.au/news/2014-07-31/ebola-timeline-deadliest-outbreak/5639060.

The date I used as “day zero,” March 25, 2014, is the day when the Ministry of Health in Guinea announced an outbreak of ebola was in progress, according to this source: https://en.wikipedia.org/wiki/Ebola_2014. It started earlier, of course, but was not widely known before that date. The last data points shown are for July 27, 2014, the most recent date for which I have the needed information.

## A Half-Solved Mystery: Rotating a Sine Wave

### Image A few minutes ago, I wondered how to write a function whose graph would be a sine curve, but one that undulated above and below the diagonal line y=x, rather than the x-axis, as is usually the case. How to accomplish such a 45 degree counterclockwise rotation?

Well, first, I abandoned degrees, set Geometer’s Sketchpad to radians, and then simply constructed plots for both y = x and y = sin(x). Next, I added them together. The result is the green curve (and equation) you see above.

This only half-solves the problem. Does it undulate above and below y=x? Yes, it does. However, if you rotate this whole thing, clockwise, one-eighth of a complete turn, so that you are looking at the green curve going along the x-axis, you’ll notice that it is not a true sine curve, but a distorted one. Why? Because it was generated by adding y-values along the original x-axis, not by a true rotation.

I’m not certain how to correct for this distortion, or otherwise solve the problem. If anyone has a suggestion, please leave it in a comment. [Note: an astute follower of this blog has now done exactly that, so I refer the reader to the comments for the rest of the story here.]

## When A Sine Wave Is a Circle

### Image When y=sin(x) is plotted on a polar coordinate system, with everything set, consistently, to radians, the resulting graph is a circle sitting atop the origin, with unit diameter.