The Bailout Rap

When is a sphere not a sphere?

Yesterday I posed the following question to a few of my more mathematically-inclined friends:

"If you have an orthographic representation of an object (that is, views from the front, top, and left sides), and from each angle it appears as a circle, is the object guaranteed to be a sphere? And if not, what other shape could it possibly be?"

Here it is again, in simpler terms.

Let's say you have a ball. If you trace the outline of that ball, it will be a circle. Take that ball and rotate it 90 degrees forward, trace the outline, rotate it 90 degrees sideways, and trace it again. Can you make any modification to that ball without affecting the outlines?

I probably wasted more than an hour yesterday trying to figure out this simple puzzle. There is one trivial solution which does not count, and that is anything affecting the invisible 1/8th of the ball.

I finally realized the solution earlier today when I glanced at the basketball on my floor. I recommend thinking about it for a while on your own, then when you're done you can find out.

No, the object is not guaranteed to be a sphere.

You can easily prove this by taping a small object to the surface of a ball, then rotating it appropriately. The key is to have the small object fall as far away from all of the "edges" as possible.

This makes sense. If you outlined the ball only once, you could put anything you wanted on top of the ball and it would still make a circle. If you rotated it and then outlined it again, you would be able to put something near the top, but not quite exactly at the top. Each time you outline the ball from a different angle, you impose more and more limitations. The only way to prove that the object is actually a sphere is to make outlines from infinitely many angles.

I'm convinced that this can be represented mathematically, but I'm just not quite sure how. Perhaps one of you is up to the task?

Update: The maximal deformation can be represented as the intersection of three cylinders. This shape is called a Steinmetz Solid.

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The end of personal blogs

I have completely neglected my blog. So has everybody else.

Of my friends with blogs, half of them have not updated in 3 months. One hasn't updated in over a year. In fact, the most recent update on any blog was a full month ago!

Blogs come and go, that's a given. People have short attention spans. But when an entire group of blogs fades out of existence, there must be something else afoot. Our personal blogs are being replaced in great quantity by structured "social networking" websites such as Facebook and microblogging services like Twitter.

The reality is that those services have provided better alternatives to achieve the goals that we were trying to achieve with blogging. For a small personal blog that only gets read by friends and family, the focus is mostly on updating those people about our personal lives. Facebook is now the most popular way to do this, and it takes much less effort than blogging.

I've spent some time thinking about how this affects me and the handful of people that read my blog. I have decided that I will not give up on blogging, but rather pursue a different direction. In the future, my posts are likely to be of a more technical nature which will reflect on the nature of my interests.

I hope that you will continue to find my posts interesting, and I look forward to reading your comments.

Relay for Life times

My time slots for Relay for Life are as follows:

May 10, 9:00 PM - 10:00PM and
May 11, 12:00 AM - 1:00 AM (yup, that's midnight!)

Feel free to join me! :D