Math Is Fun Forum

gourish

Member

Registered: 2013-05-28

Posts: 153

mirrors #2

if two mirrors are placed at angle θ to each other then find the number of images formed by an object placed in between them....
i just found that the number of images(n) is given by n=[360/θ]  for n being even and [360/θ  -1] for n being odd and [.] is the greatest integer function but i don't know how this was derived

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"The man was just too bored so he invented maths for fun"
-some wise guy

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: mirrors #2

hi gourish,

n even formula looks right to me but:

When the angle is 72, I'm getting 5 images.  To prove the property I think you'll have to analyse the angle between the red lines (see diagram) for different angles.

I can prove it for any particular angles like 90 and 60.  I cannot think of a way to prove the general result in a single proof.

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

gourish

Member

Registered: 2013-05-28

Posts: 153

Re: mirrors #2

hi bob,
i still don't get it... since the image can be placed anywhere between them i am unable to make any relation with the number of images formed can you please explain me how to analyse the diagram that you have posted i am still having trouble dealing with the resulting proof

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"The man was just too bored so he invented maths for fun"
-some wise guy

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: mirrors #2

Hi Bob

Try to do that pic with only half of the figure you posted there. I think you are assuming the object to be symmetrical and halfway between the mirrors.

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“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: mirrors #2

hi Stefy,

To start with, that's exactly what I did.  See diagram below.

I set the mirrors at 72 degrees and chose an F shape because it has no lines or rotational symmetry.  That was shape 1 on the diagram,

Reflecting in BC gives 2.

Reflecting 1 and 2 in AB gives 3 and 4.

Reflecting 4 in BC gives 5 and so on.  Successive reflections give images up to 10.

Thereafter, no further images are generated by reflecting any shape in either mirror.

So I thought ten images.  But then I realised that 1 and 7 together can be treated as a single 'motif' in which case the number of images is reduced to five.

So I moved 1 until it joined to 7 to make a single shape.

I think this happens whatever shape you start with.

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

gourish

Member

Registered: 2013-05-28

Posts: 153

Re: mirrors #2

but it can take a lot of time to find the exact number of images being formed isn't there a better way then this that i can use? other then the formula that i mentioned because it doesn't work when the mirrors are the 3 sides of an equilateral triangle....

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"The man was just too bored so he invented maths for fun"
-some wise guy

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: mirrors #2

hi gourish,

Mirrors around an equilateral triangle is  not two mirrors???

I don't know an quick way to get a formula.  Sorry. 

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob