Hey ladies (and guys with long hair) - remember the last time you brushed your hair, and saw the tangle of long fibers left over in the sink. You were tempted to try rinsing them away, but knew that you’d see them again in the form of a nasty hairball. So, you retrieved them responsibly with the closest Kleenex.

OK guys - imagine the last time you shaved, lathering up with a dallop of foam and slicing hundreds of whiskers from your cheeks with tiny steel blades. remember the sink bowl after shaving. It was full of tiny hair clippings a few milimeters in length. With a few quick splashes, they rinsed down the drain quickly along with the rapidly dissolving foam. You were out the door, knowing you’d never see those fiber fragments again.

Hey everyone - now imagine if you tried to rinse the same amount of whisker bits down the sink, but as longer strands. In other words, pretend that you’ve pieced those strands together to form a smaller number of longer hairs. At one end of the scale, there is a single hair equal to the length of all individual shavings. At the other end, there is a large number of tiny, powder-like hairs chopped into minimal lengths. The drain opening is a perforated disc of circular holes, which acts as sieve to prevent large objects from passing through.

At what point do tiny hairs become a clog? With enough soapy hot water, an unlimited amount of near-powder hairs can pass through the disc. However, no amount of rinsing can push a nasty hairball of long fibers through the disc.

The essential question:

What length, between the shortest possible fibers and longest possible fiber, will stop fibers from rinsing through?

Assumptions:

• All individual fibers are uniform in length, thickness and flexibility.
• Fibers will only bend minimally – they won’t curl up into a ball to pass through a drain hole.
• The fiber diameter is much smaller than the drain hole diameters.
• Fibers can pass through the drain holes in any orientation.
• An infinite supply of hot, soapy water is provided for rinsing.
• The drain can block fibers. Other fibers can also block fibers, and so on.
drain

#####
##     ##
#         #
#           #
#   ##        #
#  #  #   ##  #
#   #  #  #  #  #
#    ##   #  #  #
#          ##   #
#   ##          #
#  #  #  ##     #
#  #  # #  #    #
#  ##  #  #   #
#       ##    #
#           #
#         #
##     ##
#####

shortest fibers (near powder)
- - - - - - - - - - - - - - - -

shorter length fibers
-- -- -- -- -- -- -- --

longer length fibers
---- ---- ---- ----

longest fiber (single fiber)
----------------

When the length of individual fibers is equal to or less than the diameter of individual drain holes, they will pass through easily. When each fiber length is much greater than the drain hole diameters, they will be blocked and no amount of water will rinse them past. However, some fiber length that is slightly longer than the hole diameter will also be able to pass through. Fibers can thread themselves through the holes by passing a first end through a hole, and then following through.

Fibers can also pass through by flexing. When placed lengthwise over an opening, they can bow inwards and get pushed through due to the force of passing water. Also, a combination of flexing and threading can send a fiber through a hole. For example, the first end of a fiber can bow inwards and pass through the hole, followed by the rest of the fiber.

So when is a heap not a heap? When is a clog not a clog? When does a quick shave turn into an emergency call to the plumber?

This seems to be related to other biological phenomena, especially arteriosclerosis and blood coagulation. For each of the sink drain, clogged arteries and healing wounds, there are fibrous elements (either hairs or platelets) and a lumen that the fibers need to traverse (a drain sieve or the lumen of a blood vessel). The fibrous elements can be blocked by both the lumen size, or by other fibers that have already been blocked by the lumen.

What point separates moderately clogged arteries and a heart attack or stroke? When is the time to step on a treadmill for a half hour versus scheduling surgery? What point separates clot-ready blood and a healed wound?

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