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Monday, January 1, 2018

Winter fable : ant and grasshoppers ~ anew !!!

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) 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.  (

With regards – S. Sampathkumar
27th Dec 2017.

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