On The Road ... Map

On a trip to visit family, we stopped at a gas station in Hammondville, Alabama. On the wall was a map of the US with this wear pattern of customers touching where they were on the map. The many touchers have worn though the paper map down to the underlying supporting board. It seems that many have traced their path of travel extending southwest to Birmingham, AL and northwest to Chattanooga, TN likely along the connecting route US 11, passing through Hammondville, or along the parallel interstate 59 a bit further west. What remains is a roughly ellipsoidal bivariate frequency distribution of wear with a greater frequency of wear centered on Hammondville and lesser frequencies of wear in ellipsoidal contours around it.

Wearing of the Sphere

This is a kids' playground item at Wheaton Regional Park in Maryland. It's a toy/chair/apparatus that is basically a painted sphere on a pedestal. Children sit, slip off, or perhaps spin around, balancing on top of the sphere. This produces the wear pattern contours shown here. The central circular region has the most paint rubbed off, down to the metal interior. Around this is a ring of lesser wear. The circular nature of the contours indicates that the children don't seem favor one approach to sit, one method of sitting, or one direction of dismounting the sphere. No one direction of wear seems overly favored, or avoided, even though the light ring of lesser wear may not have exactly equal width around the contour. What remains are the circular contours of the frequency of use of a distribution defined on a sphere.

Stepping Up

This is an image of the floor in front of the checkout counter at a deli in Rockville, Maryland. Standing and awaiting completion of their purchases, customers have worn away the floor tiles with their shuffling feet, revealing contours of their use. The central brown area shows the concentrated region of the most wear. So frequently has this central spot served customers that the tile has been worn through to its interior, well deeper than its decorative surface. Around that, lesser use has worn off only the pattern of the floor tile. This leaves a white annulus of wear surrounding the central region. Beyond this there is a dirtier area of the floor tile where customers queue up to pay for their purchases. This all results in an image of the somewhat circular contours of the joint frequency distribution of customer foot placement at the checkout counter.

Bang the Drum Differently?

Here is another snare drum pattern of wear showing the joint, two-dimensional pattern of drumstick or brush hits on the head of the drum. Again we can see the central dark and quite circular contour of the region of most wear. This is surrounded by a more irregular discoloring of the drum head, likely from the metal brushes we see here. This is very different from the nearly circular pattern of snare drum wear we have seen earlier. Is this the results of different drummers and their particular style or of similar style drummers playing different types of music? Or perhaps both?

Sobriety Contours

Pattern from the joint distribution of keyhole use labeled with contours of sobriety (thanks Scott) from this source (among many). Perhaps this owner needs the keyhole for drunk people.

Normal Snare Distribution

Here is the head of a snare drum (thanks Sean) showing the two dimensional, joint distribution of his drumstick hits. The maker, Evans, produces drum heads with two plys of plastic bonded together. In this image the pattern of wear and use reveal themselves through contour lines as closed curves indicating regions of similar frequency of use. The greatest wear is in the lightest colored central region, having seen the greatest frequency of hits. This central region has worn through the first ply showing the remaining plastic support underneath. Sean seems to have a very stable left hand, consistently hitting this small central region in nearly a circular pattern. In this region, the horizontal location of his hits seems independent of the vertical, producing this near circular pattern of the joint distribution. Surrounding this is the darker region of the top ply of plastic. This layer retains more dirt and grime than the underlying supporting plastic. Again the pattern of these hits is nearly circular. It turns out that with simple assumptions, like radial symmetry and independence, the pattern can be shown to be that of a bivariate normal distribution. A result that was thought first published (p. 398) by John Herschel in 1850, but actually discovered much earlier by the American mathematician Robert Adrain in 1808. More details on that in a future post.

But here, perhaps we see slight deviations from normality. The darker ring seems to show slightly  more variability extending vertically and a greater clustering of hits on the bottom of this image. This indicates a bit of skewness towards the top of the picture. Less use and wear is finally shown in the cream colored outer region that has seem very few hits. Thanks again Sean.