Thursday, 26 December 2019

Beware of insects with moustaches


Water must flow for all
I have written about my obsession with showerheads before, the way I tease the rubberised nipples if it is restricted or no water flow from any of the teats, this usually due to calcification.
It is not enough to stand under the shower, I look up and if just one of the teats is restricted, it needs to be sorted out. Every teat must be supplying a spray of water at its optimal capacity. In our apartment, I was met with a little extra difficulty, the teasing of the nipples did not work on two of the outlets of the innermost circle of 5.
The overhead shower which is a Hans Grohe product, the company founded in 1901 and is probably a global leader in faucets, showers, and taps with the model ‘Raindance’ of which there are many variants.
A census of the teats
After a few trials, I attempted unpicking the teats with a toothpick to no avail. Eventually, I dismantled the showerhead and found calcified deposits in the feeder assembly that I was able to unpick with a toothpick. On reassembling the showerhead, all the teats began working as intended.
Another thing that had bothered me was finding a simple mathematical formula to calculate the number of teats on a showerhead. There are diverse types, square, circular, rectangular, oblong and so on. This determines how the teats are arranged in the showerhead.
I have mostly encountered the circular arrangement with concentric circles with a quintuple setting in the innermost circle radiating out to 6 or 7 concentric circles. On the basic count, there were 5 on the innermost circle, then 10 on the next and 15 on the following.
A formula is reused
This would suggest an arithmetic or mathematical series. So, if there were 3 concentric circles based on multiples of 5 teats in each concentric circle, the total number of teats would be 5 + 10 + 15 = 30.
What makes this interesting is the cumulative number of teats and the series developing. On the 1st circle it is 5 or 5 * 1, for the second it is 15 or 5 * 3 and on the third, it is 30 or 5 * 6. I well-known series of 1, 3, 6, 10, 15, 21, 28 … is forming in the process. This, I have learnt is the triangular number sequence.
The detail from the link about suggests a formula of n(n + 1)/2. This would deal the determining the number of teats for regular concentric circles where each consecutive circle is a multiple in the natural sequence of the innermost circle. The series in the paragraph above would suffice for where the innermost circle has one teat and the next concentric circle has 3 using the triangular number sequence.
Hansgrohe Raindance showerhead
All numbers matter
The showerhead in the picture has m = 5 teats on the innermost circle with n = 6 concentric circles. For which I now have the formula m * n(n + 1)/2 and whilst the formula can be decomposed further, it is neater to keep it this way. I would then have 5 * 6(6 +1)/2 = 5 * 6 * 7 / 2 = 105 teats.
On the referenced blog where the rainfall showerhead at the Royal York Hotel had 7 concentric circles with 5 being the multiple, the applied formula would result in 5 * 7 * 8 /2 = 140 teats.
With my showerhead conundrum solved, I will happily bother my head with something obscure and productively silly as finding out if there are really any insects with moustaches in Cape Town, this showerhead mystery already fits the bill.
 Courtesy of the William Kentridge exhibition at Zeitz MOCAA, Cape Town.



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