In Mathematics, there is an interesting puzzle : ‘the grasshopper Q’: A grasshopper lands at a random point on a
lawn, then jumps once, a fixed distance, in a random direction. What shape
should the lawn be in order to maximize the chance that the grasshopper stays
on the lawn after jumping?
Cricket is a grasshopper; Grasshoppers are insects of the
suborder Caelifera within the order Orthoptera, which includes crickets and
their allies in the other suborder Ensifera. Grasshoppers are typically ground-dwelling
insects with powerful hind legs which enable them to escape from threats by
leaping vigorously. They hatch from an egg into a nymph or "hopper"
which undergoes five moults, becoming more similar to the adult insect at each
developmental stage. These insects are plant-eaters,
with a few species at times becoming serious pests of cereals, vegetables and
pasture, especially when they swarm in their millions as locusts and destroy
crops over wide areas. In
another, a grasshopper is a sweet, mint-flavored, after-dinner drink. The name
of the drink derives from its green colour, which comes from crème de menthe.
From
childhood we have heard and read the fable of ‘grasshopper and ant’ - of a grasshopper whiles away the entire
summer months singing while the ant works diligently stocking up food for winter. When winter
season comes, the grasshopper finds itself dying of starvation and begs the ant
for food. To its reply when asked that it had sung all summer, it is rebuked
for its idleness and advised to dance during the winter. It is a moral story on the virtues of hard
work and planning for future. In life
too, one gets easily tricked into false sense of security believing that all
things are in place ~ yet life can be different and difficult, when some
changes take place – in earlier ages, it was ‘what after retirement’; now the Q
that beckons is ‘when retirement’? – whether it is by choice or forced and what
after such a break ? Pink slips are
often heard in market place and sometimes big firms too ruthlessly and
mercilessly reduce the size, sending people overnight.
That
places significance on financial planning, career planning and pension
planning. There is no single right
solution on how much and when – but there is always the right answer that one
must save and plan their future.
Some
years back there was an article in Financial Times that added another
perspective to ant and grosshopper. The
latter is lazy while the former piles up huge stock for its future
(winter)..life can be much more complicated than the fable scenario. It likened the ants with Germans, Chinese and Japanese, while the
grasshoppers were American, British, Greek, Irish and Spanish. Ants produce
enticing goods grasshoppers want to buy.
The worker ants produce goods, while the lazy grasshoppers take them on
loan. Being frugal and cautious, the ants deposit
their surplus earnings in supposedly safe banks, which relend to grasshoppers.
The latter, in turn, no longer need to make goods, since ants supply them so
cheaply. But ants do not sell them houses, shopping malls or offices. So
grasshoppers make these, instead.
When
economy fails and there is chaos, grasshoppers suffers and as their purchasing
power vanish, ants too suffer. Eventually
when the loanees go bankrupt, the lenders too suffer seeing their hard earned
money vanish in thin air – that creates ant nests in poor countries and rich
nests in richer countries. Then another
clever merchant country emerges by exploiting the economic situation. The moral is – no useful purpose served in
simply saving and more so, in lending to people from whom it is not going to
return and making that decision is always difficult.
Away
from the fable, the factual position according to entomologists is seemingly
different. An entomologist at Washington
State University explains that in places
with colder winters, such as Washington state, grasshoppers spend the winter as
eggs. That means that their mothers will have buried them deep in the
ground.The grasshopper mom has an egg-laying organ, called an ovipositor,
that’s shaped like a knife or sword. It’s really handy for digging in the
soil.Some Pacific Northwest grasshoppers, like the red-legged grasshopper, will
lay about 20 eggs at once. The mother will cover them all with a gummy
coating.The coating hardens and binds the eggs together so they can survive the
harsh winter conditions. The mother grasshoppers will also bury them.“They
hatch in the spring when the weather warms up and the sun comes out.” “In warm
places, grasshoppers are more active in the winter because the temperature is
good and there are plenty of plants around to eat.
So
grasshoppers may not go to ant seeking the saved food !
If you still remember the Q at the start :A first impression may
be that the lawn should be in the shape of a circle, at least when the distance
the grasshopper jumps is small. However, Olga Goulko and Adrian Kent, the two
physicists who introduced the grasshopper problem in a new paper, have
mathematically proved that a disc-shaped lawn is not optimal for any
distance.Instead, they discovered through numerical simulations that the
optimal lawn shape takes on a variety of complex shapes for different jumping
distances, such as a cogwheel shape for distances smaller than 1/π1/2 (the radius
of a circle of area 1, or approximately 0.56), while for larger distances, the
optimal lawn consists of disconnected pieces. Often, but not always, these
optimal shapes possess some type of symmetry.
(www.phys.org)
With
regards – S. Sampathkumar
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