Las Meninas - Science of Perspective

Be it in the laboratory of a scientist or studio of an artist, similar processes are at work. They both possess an eye for detail. An artist's keen sense of perspective is critical for capturing on canvas the image of the eye.

Perspective is one technique that can make artist's work seem real, give depth and also make people go wow. Early 13th and 14th century artists used lines that splayed rather than converge when they approached horizontal lines. This could be considered a major step towards the beginning of perspective drawing. Real change came during the renaissance period with the invention of linear perspective.

Linear perspective is a mathematical system of projection of three dimensional object on two dimensional surface. You begin with, say a horizon line which is effectively defining the farthest distance of your background and a central vanishing point. The foreground of the space is the bottom of the picture plane, from here orthogonals are drawn to the vanishing point. All these elements (horizon line, vanishing point and orthogonals) establish the space of the artist. The figures and object drawn in this space appear to exist in 3D.

Suppose the artist want to further complicate the space, for example in representation of square tiled floor. He chooses distance point on the horizon line and connect the bottom of picture plane through the orthogonals. The points at which orthogonals bisect the line establishes horizontal lines called transversals. these lines perspectively represents the square tiles in space

This is the sweet spot where we find the link between the science of 3-D geometry and illusionstic representation

But all this we read above is not enough to analyse the masterpiece Las Meninas. The beauty of this piece is in its numerous elements and spotting the perspectives involved.


Diego Velázquez’s was an artist in the court of King Philip IV of Spain. He painted numerous paintings, but this continues to remain the most remarkable painting of all.

I suppose, in the first glance it doesn't seem that exemplary at all.


Being a court painter it isn't surprising that Velásquez choose the scene at palace to display his skill set and experience.



There are 11 characters in his painting. The focus of this picture immediately falls on the little princess Infanta Margarita Teresa.
She seems to be geometric centre of the canvas. She is illuminated by the light from the window on right.
Apart from the princess there is an obvious pairing of the characters in the canvas.Like the two dwarfs (4 &5), the chaperones(6 &7), the maid and the man at the door way, the king and the queen on the mirror(10 &11), finally the maid and Velázquez himself (3 &9).
So these pairs accentuate princess as the focus of the painting.

But this is where Velázquez takes perspective to another level. There are multiple weights of focus on his painting. Its not just on the princess.

What elevates this painting, as much as it confuses, is the mirror that forms the space between background and foreground. The mirror reflects the Royalty that is King Philip and Queen Mariana. It takes the viewer to a space outside of the painting. But this where the unresolved mystery lies. The reflections on the mirror, will put the viewer in the feet of royalty. But closer analysis leads us to conclude the actual vanishing point for the viewer is the door way in the background.




These multiple focal points increases the viewing experience. There are  questions regarding the light reflecting of the mirror. Is the mirror reflecting the canvas Velazquez is working on? If it is, then where is the actual painting of King Philip and Queen Mariana, which apparently doesn't exist because Velásquez never painted one.








But on further analysis, Velázquez is looking directly at us, the viewer. This means we become the mirror plane. The entire scene could be reflecting of us. This adds new perspective to his painting. There was tradition that existed about painting a portrait where a large mirror is placed in front of the subject and the artist paints the reflections. The size of the actual canvas is large and just like the canvas on the the painting. The entire mirror could be the viewing plane.






As much as we see the focus on royalty, that is the princess, King and Queen, the surprising focus on the doorway is undeniable. In fact we can see the illuminated rectangle doorway is in line with the mirror on the same plane.

The significance of this has been debated and said to be reflecting historical context at the time of the painting.
The healthy rosy princess depicting ideal world, the doorway the real world and the mirror the reflections.



The viewing experience is beyond the canvas. The perspective enables the viewers to walk through the canvas.

This painting is one of the most analysed painting in the history. Its analysis of its perspective makes it an intellectual endeavour and takes the painting beyond just a work or art. It most definitely, according to me are the strokes of a mastermind, the best snapshot.

 There are plenty of books, and articles regarding this painting.

This brings me back to then why I wrote this article. It all started when I read about one of  my favourite Physicists Richard P Feynman in one of his anecdotes regarding an argument with artist. He told the story of his disagreement with an artist about who can better appreciate the beauty of a flower: artists or scientists. While his artist friend thought ripping apart the flower retracts its beauty, Feynman thought knowing the processes behind the beauty of the flower could infact enhance the actual beauty.

But in this ironic example which as a "work of art" just looks plain on casual observation and like any other painting of the century. But analyse its close elements, break it down and what you see evolving is really the "Science of art". 

Do check out

• For more on Las meninas which was one of my references on this subject

• http://secrethistoryofart.blogspot.in/2010/12/velazquez-las-meninas.html
• http://mentalfloss.com/article/68209/14-things-know-about-velazquezs-las-meninas 

P.S : Dedicated to my artist friend. Also do post me your "perspective" about the "Las Meninas" 

WITHOUT NUMBERS (Part 2 - Peano)

WITHOUT NUMBERS (Part 2 - Peano's Axioms)

For Part 1-WITHOUT NUMBERS (Part 1- The horror)

Thus ended the narrative of the mathematics lecturer who had come as substitute for a day. The professor managed to catch the attention of the class in the dreadful afternoon hour.  There was pin drop silence that followed his story. He went on to say that we should live the story we want to tell.  This is the story of numbers and how mathematics evolves its structure.

We were all like


He went on to say that Mathematics, in its essence, is a subject in which one begins with a set of concepts and rules and then rigorously works out their precise consequence.

Thus began an interesting class..

First of all I am sure you all have the same question
 What are natural numbers then?

The informal definition your textbooks  will give 
"A Natural number is any element of set N= {0,1,2,3....} ( We often call the set containing zero and natural numbers as whole numbers. But it is fine here)
which is the set of numbers created by starting with 0 and counting forward indefinitely"

Before we could note that down.

Our Professor...





This definition doesn't seem so satisfactory because then the question is what is counting forward indefinitely?

Intuitively the definition is okay, but we all knew from the story (see Part 1-WITHOUT NUMBERS (Part 1- The horror) our intuitions are also not entirely acceptable; for instance how do we know it is possible to keep counting indefinitely without cycling back to 0? More importantly how do we perform operations like addition, Multiplication, exponentiation and so on ?

One standard way of doing it is in terms of Peano's Axioms. The idea is to just present a complete set of axioms to get N 

and Nothing is taken for granted or treated as obvious.

The intuitive picture is N consists of 0 and everything can be obtained from 0 by incrementing. So our beginning is in the existence  of 0 and an operation called incrementing.

Axiom 1: There exists a natural number denoted by 0
We can imagine emptiness or nothingness. We denote 0' the successor of 0 and 0'' as successor of 0' that is successor of successor of 0. We also use usual notation 1=0', 2=0'',...

Axiom 2 : If n is a natural number then n' is also a natural number
0 is a natural number we know from axiom 1, 
0' (1) is a natural number by axiom 2
0'' (2) is a natural number since 1 is a natural number
0''' (3) is a natural number since 2 is a natural number

Professor stopped and asked us to consider if these two Axioms are enough to define N?

This time we did identify the problem that is the above definitions do not rule out the possibility of 0 being a successor of a natural number.
Ex; consider the set{0,1,2} where 0'=1, 1'=2 , 2'=0. It does satisfy axiom 1 and 2 

Quick fix 
Axiom 3 : 0 is NOT the successor of any natural number

Oh we should be done now...

But consider {0,1,2,3} and 0'=1, 1'= 2, 2'= 3, 3'=1 

The successor should not wrap back to the earlier natural number

He was definitely pushing us to the limit where we all did go like..



 
Then we need

Axiom4: If n, m are natural number such that n'=m', then n=m or equivalently if  (the symbol != will be used to represent "not equal to") n != m, then n' != m'. This means different natural numbers must have different successors.

ex Prove 5 != 2  ( 0''''' != 0'')
if       5=2
---->  4'= 1'
---->  4= 1 ( by Axiom 4)
---->  3'=0'
---->  3 = 0 ( by Axiom 4)
---->  2' = 0
wait a minute that is not possible because it would then be a contradiction to Axiom 2 and 3

Hence 5 cannot be equal to 2

Our lecturer therefore making it the best class in long time concluded
These four axioms seems to give us all the natural numbers distinct from each other.

While we were all satisfied this one student didn't seem particularly convinced it is complete and said that these axioms 1 to 4 do gives 1,2,3,4 ... in set N but none of these axioms rule out the possibility of the other elements which we do not want example {0, 0.5, 1, 1.4,...}

The lecturer look stunned at the boy's observation and tell us all

It is true these Axioms do not rule out the possibility of entirely different natural numbers represented by 0.5 and then satisfying axiom 1 to 4 such that
0=0.5, 1.5=(0.5)', 2.5 = (1.5)' ... By now you have realised 1, 2, 3 are just symbols. Our interest lies in its structure.

To rule out this possibility, a very ingenious axiom, not very intuitive immediately, axiom is formulated

Axiom 5: (Based  on principle of mathematical induction) 
Let P(n) be a "property" pertaining to a natural number n. Suppose that P(0) is true and suppose whenever P(n) is true, P(n') is also true. Then P(n) is true for all n.

The Axiom 5 is in fact very very deep and really ingeneous.
 The term property looks a little vague but think of P(n) being related to property " n is even"  , n is equal to 3. Ofcourse these examples are still vague as we have not defined concepts yet. But once we define we rule out the possibility of other elements creeping in.

These are in fact Peano's five Axioms.
Thus the hour came an end and we had a wonderful math hour. The substitute lecturer was impressed with that one student's insight and asked him his name.

The student said " Joiston Reich sir"

That was the last we saw of our substitute lecturer.

Check them out (These were the links of inspiration)
A day without math 

Lots of young kid's stories on how their day would be without math
http://www.globalclassroom.org/nonumb.html


Books referred 

• Analysis 1, 2 by Terence Tao, Hindustan Book Agency, 2006
• Claude W Burril, Foundations of real number, McGraw-Hill(1967)

   P.S. Do open in web version to vote for the usefulness of the article 

WITHOUT NUMBERS( PART 1-The horror)

All through the ages, no intellectual endeavour has been more important to those studying physical science than has the field of mathematics. But mathematics is often believed to be a tool or language for science. It is also an end in itself, and as such, it has over centuries affected our world view in its own right.

 Numbers are one such invention of mankind. But I have often wondered what would it be if we didn't have numbers. If we didn't invent it at all. How different would the world be? Looking at my clock I see numbers. Hence if I didn't have numbers, will I be able to know time. When I say not knowing numbers, it means not able to understand the concept of "value" and not being able to associate yourself to words like "larger than" or " smaller than ". Then I am forced to rethink " Are numbers really an invention or a discovery?". That's when this happened...

 Many years ago, I used to teach math to a class of grade one students. Among all the kids I taught in my life, this one student I will always remember. The memory still sends chills down my spine.

 He was the young Joiston Reich. He was older than the rest and completely isolated himself from the class. He had difficulties doing his work on time. While other students took part in fun activities conducted, he sat alone staring into space. He always seemed terrified.

 I have always seen kids with particular dislike towards math but this was different. His eyes didn't flash of terror due to lack of comprehension rather it displayed apprehension and disbelief. He was always confused and lost.

One particular chilly winter day, all my students covered in their new winter clothing, sat waiting for their parents to come and pick them up for the christmas break. I see Joiston doesn't have his muffler or woollen clothing.

Feeling sorry I go up to him and ask "Son, are you feeling cold ?"

He looks up at me. That look was so frightful. The look was colder than that of winter's cold. I took a step back. He was now looking beyond me into space contemplating. I decide it is wise to let him be. I smile and wondered why this ordinary school boy was difficult.

Breaking the ice, he suddenly asks " Excuse me, (long pause) What are numbers? Are they real?"

Puzzled me " Well,(longer pause) numbers are something we created to count to see how much of something or how little of something we have and.."

He - " So they exist in our minds. We created something that exists in our minds? They are not the reality of the world right ? Just like Hogwarts or unicorns or tooth-fairies?"

Surprised that this boy is asking questions beyond his age and maybe beyond my wisdom.

I reply slowly " But numbers are everyone's reality. True you don't see them like rest of the fictional stuff. But they are not arbitrary products of one person's imagination. They are as real as you and me."

That moment Joiston was no longer a kid.
 He told me " I don't have that reality. When you tell me "that is bigger than this", I don't understand. I don't see the minutes tick nor do I feel it. I can count but I only know them as words not as value you give them. I think they are like colours and I am blind. They tell you about the beauty of the world that is meant to be extracted. I am not aware of my age. i am not aware of the beginning or the end. Is it early or too late. Is it hot or cold. I never know if it is question or questions. When to say a lot or when it is too little. I don't even know what is when. The reality of number is not something we are born with that is why we are all taught how to count. But as much as we are taught, we are also born seeing how it is real and engraved in our intuition. Hence I don't think its just simple creation of mankind, it is something that was waiting to be found. But it is not real for me. I must be fictional then."

He stops, looks at the clock in the hope to see something for the first time and then walks towards the door leaving me shell shocked. That is the last time I saw him.

To be continued...               

Science Hacks - How to impress your date (highly recommended)

Out on a date? Are you a geek with spectacles? All your life you have been fascinated with science but suddenly a human struck your fancy. Lucky as you can get he/she agrees to go out with you. As much as you like this person, you want to show how much of what you do is relevant to her and the world (and also look cool). Here are few hacks to being nerdy and fascinating. 
No, its not about dressing well and talking boring, usual stuff. This your chance to let the personality shine.
So here are some unconventional conversation starters. They will definitely exhibit your brilliant and thoughtful nature along with the quirky weird scientist you love a lot. These could be used to your advantage to enhance the date.

Since you are a scientist I am pretty sure you will the enjoy the science of subtlety.  
  
Though the following are written keeping in mind my friend who needed some date ideas and conversation starters. All the women out there, these are for you too.Try them and see them get swooned .

THE STICKY NOTE HERO  
Lets assume she meets you in the library and asks you for help. You suggest flashcard or sticky notes are the best way to get the work done on time. As luck may have it she takes out her post it, peels the note and starts writing on it.

She sticks it on her study board. This is the moment you swoop in like nerd hero and say  "Oh now I understand why you are having hard time remembering what you learnt". She looks at you puzzled.

You tell her "These sticky notes don't last that long".
She tell you true and complains about how her sticky notes keep falling from the walls and study boards.

You confidently assure her "There is way it wont fall".

Well all this needs is some demonstration. So to demonstrate, I took my sticky note pad (and I am sure you will too).  Lets call the side with the strip of adhesive, the top side. Now most of us peel the sticky note from bottom to top as shown.

   
                  Peeling it like this will cause the curling effect as shown

I did try this with different sizes of sticky notes to prove my point (and incase she carries different type of sticky note) and as you can see this kind of peeling leaves similar curling.

When you peel like this, sticky notes don't stick on the wall for long enough to remind you of the work scribbled in yesterday (Finally an explanation for her so called procrastination).
Now for the trick part

You tell her to peel the sticky note along the adhesive that is from left to right (or vice versa)

I tried this way of peeling with other sticky notes and the outcome is pretty evident.

Obviously she would want to know why does the sticky note sticks better in the latter case?

Macroscopic representation of microfolds 

Well now to reveal the best part of the story (the nerd in you is ecstatic). You tell her peeling the sticky note from bottom to top creates microfolds that are parallel to the top side as shown. This causes the curling.

Macroscopic representation of microfolds

But on the other hand, the microfolds created while peeling it along the adhesive band are perpendicular to the adhesive which reduces the curling effect due to better spacing.
Clearly don't try telling that in one breath. Be sure to tell her in the most amusing manner possible because this piece of information will blow her mind. It may seem silly to talk about sticky notes but this may ensure that she sticks around you longer.
Who doesn't like the guy with solution to her tiniest problems




 DOWNED POWER LINES DRAMA

You are walking down the lane beside her after a rainy day and lets assume your chances of encountering a downed power line is pretty high.

If you spot a downed power line walk the other way immediately with her and you call the police immediately.

Now she is bewildered at your dramatic behaviour on downed electric wire. This is the moment you show how much you care.

You tell her even walking near a downed line can electrocute you because the voltages are high enough to push electricity through the dirt nearby (pause) and that could kill us.
She does blush and asks you even more dramatic question "Now what do you do if a power line falls down near us?"
Then you assure her there are still ways to ensure no harm is caused (macho man). She looks at you enquiringly.

You tell her "I would immediately stop you from moving, ask you to jump straight in the air and land with your feet together. Continue jumping away from the line, keeping your feet together, until you're at least 50' away".

She replies " I would look silly and stupid doing that almost like a kangaroo".

Well the hero nerd in you swoops in again and tells "Yes you may look silly and adorable doing it. But definitely not stupid because this is the concept of step potential, where one point on the ground is at a higher voltage than another. If you walk normally, your left foot may be several hundred volts higher than your right which would cause electric current to flow through your body and kill you, which I am not ready to risk. Keeping you feet together keeps it in same potential and prevents current from flowing."


You could tell her that in one breath and definitely that will take her breath away.



TOOTHPICK TRICK

Well you go on a dinner date to high end restaurant. You have nothing interesting to say. You are done with the usual set of comfortable topics. Now she asks you what do you do. You tell her you do physics. You can see the interest fizzling out.
Suddenly with twinkle in your eye you ask her "Do you find the toothpick interesting?".

She looks at you like she is having bad date and about to run but she tells you "Not really. Its just used to pick out the dirt in your teeth. What can possibly be interesting about it".

Then you swoop in like the genius hero and tell her "Well do you see the ridged end of the toothpick. Ever wondered why its there."


She looks at you puzzled but interest ignited because what possibly can be interesting about that.

You tell her "In order to correctly use this toothpick, you must first break off this ridged or patterned end. Then use the remaining needle portion as toothpick like you normally do. Once you have used it, simply place the ‘needle end’ on top of the broken-off ridge. This prevents the tip of the toothpick from getting contaminated by coming in contact with any other surface. This way, a single toothpick must last for one complete meal. It can be used multiple times."

She is like "Ohh I didnt know that. That is very interesting."

You tell her "Just like that if you get to know what I do, you may find it interesting too."

Baam!! That is confidence and pretty impressive.



Being all stiff, formal and having usual conversations during a date is safe but also boring. So successful date is a date that will be remembered and these dates have quirky moments. So being a little geeky, funny and downright scientific makes you all the more interesting. So go ahead and have a wonderful date(do tell me about it).

Well if it works great or else...or else stick to

 The content help was provided by the following. Check them out

• Sticky note ideas blog.whitehorses
•  Toothpick Digging Into the History of the Humble Toothpick
• Concept Of step potential http://stepandtouch.com/whatis/
• Quora 
• Google Images