Fair, Minimal Symmetry Dice

Mathematician Henry Segerman demonstrates his 3D printed skewed or squished six-sided dice. He states "they work like ordinary dice". He exploits 3D symmetries to produce these isohedral dice with "just enough symmetry to be fair". Check out his video above and links to his extra footage.

Menu Basket Queue

This is a view of a basket of menus at a restaurant in Chincoteague, Virginia. Notice the pattern of marks left as groups of menus scratch the wall when they are returned to the basket. As customers are seated at the restaurant, they are given menus that are removed from the right hand side of the basket. After ordering, the menus are returned to the basket and placed to the right of the remaining menus. When a single is menu returned, it nicks the wall at a location that depends on how many menus are currently in the hands of customers. If few menus are out with the customers, more remain in the basket and the wall marks of this returning menu will be further to the right. If many menus are out with customers, say just before the lunch rush, this returning menu will make a mark on the wall further to the left.

But it is not often that a single menu is returned alone. It is much more likely that a group of menus will be returned to the basket in a bunch. The size of the bunch that is returned is random depending on the size of the party seated. Each of the menus in the bunch makes a mark on the wall as they are returned to the basket. What we see is a steady state distribution of the number of seated customers, with menus in hand, waiting for their order to be taken.

Manhattan Metric 2

In the previous post we saw use of the program Galton that maps out on city streets how far you can travel in 10 or 20 minutes. Displayed on a rectangular array of streets and avenues, square or rectangular regions develop, as walking is constrained to follow the paths of the gridded streets.

The image above is a Google Earth view of the parking lot of an office building in Maryland. Commuters have parked their cars to enter a building just off the image at the lower left. They must follow perpendicular paths and walk between the cars and/or along the lanes to enter the building. But to minimize the distance of the walk, most have parked along lines of equal distance from the bottom left according to the Manhattan or city block metric. A few stragglers don't fit this pattern, perhaps wanting to protect their cars from door dings or just get a little extra exercise. But the prominent pattern in the image above is one quadrant of the rectangular 'circle' of the city block metric.

 This line graphic from Taxicab Geometry.

Manhattan Metric

Urbica is a design firm specializing in urban data analysis. They have developed a program called Galton that graphs, for a few select cities, how far you could walk in 10 minutes (in dark blue) or 20 minutes (in lighter blue). The map of Manhattan above shows those regions for a walk originating at Broadway and 42st Street. As you walk NYC you are, for the most part, constrained to travel the grid of avenues and streets. Of course, you cannot travel as the crow flies. If you could, these regions would be concentric circles with a perimeter an equal (Euclidean) distance from your start. But walking the streets, your distance is measured by the city block metric (also known as the taxicab metric or more appropriate here the Manhattan metric). This measures distances constrained along perpendicular avenues and streets. Plotting points of equal distance with this metric from would result in the roughly rectangular (or diamond-shaped) regions shown above. Since the streets and avenues are not equally spaced and obstacles can block our travel, we don't see perfect square or rectangular regions. By the Manhattan metric, circles become squares.
Via Maps Mania.

Next week, we will see directly the results of minimizing the distance traveled in such constrained walking.

Wow, It's Hot

From xkcd, a timeline of the earth's temperature from 20,000 BCE to the present, best seen here.

When people say "the climate has changed before" these are the kind of changes they're talking about.

Since the last ice age glaciation, at no time has the climate changed faster than in the last few generations!

Maybe This Information is Beautiful, It's Just Not Accurate

This is an image from the e-book Information is Beautiful from the site of the same name. But they've made the same mistake that has long served to help people learn How to Lie with Statistics by Darrell Huff.  The graphic below, from Huff's book, shows the comparison of bags of money representing an average weekly wages from two countries. The one on the right earning twice as much as the one on the left. Showing how to lie, the bags are drawn so that the on the right is twice as tall as the one on the left, but this doesn't give us the correct visual image, since the graphic artist has doubled both the height and the width to produce the image on the right. The resulting visual impression is that the bag on the right is four times the one on the right. A perfect image to mislead.

Now the graphic from Information is Beautiful purports to represent the percentage of children in poverty. On the left is shown a small shadow silhouette of a young child with arms raised that represents 2%, the percentage of children in poverty in Denmark. Compare this with the larger silhouette for Germany representing 10%. Even in this flat 2-d outline more than 5 of the Denmark outlines could fit in the Germany outline.  The problem, of course, is that a graphic artist has lied again and doubled both the height and width of the silhouettes to represent these numbers. This distorts any comparisons that could be made with these data. And it gets worse, since we are to understand these are images of 3-dimensional children!

I recall decades ago this, now classic graphic, when it appeared in the Washington Post. The same mistake of displaying data using the same error with the same methods (compare the Eisenhower dollar with the Carter dollar). You would think....

Location is Time or Money

Using data from over 175,000 rentals from the real estate listing service StreetEasy, the site FiveThirtyEight asks the question:

How much would you be willing to pay to shave a minute off your commute? For New Yorkers, the answer appears to be around $56 per month. That’s how much more New Yorkers pay in rent, on average, for a one-bedroom apartment that’s a minute closer by subway to Manhattan’s main business districts.

They plot median monthly rental versus commuting time to the 42nd Street and Chambers Street subway stations. They fit, what appear to be non-parametric regressions, curves to four different groups of rentals: studios, and 1- ,2- , and 3-bedroom apartments. As expected, rental prices fall with a longer commute. From the 1-bedroom curve they estimate that, on average, a one minute shorter commute costs around $56 more a month.