The 100th Objectivity

I'm really pleased that today we've posted the 100th episode of Objectivity - and it's a special object indeed.

You may have seen in the news last week that a first edition of Issac Newton's Principia Mathematica sold for £3.7 million.

Well in our latest video have another first edition (a better one in my opinion, because of who owned it).

But we also have THE ORIGINAL MANUSCRIPT - handwritten with notes and used by the printers.

On the same day we posted episode 100, the channel also reached the milestone of 100,000 subscribers.

Thanks to everyone who subscribes, and if you do not subscribe then here is a link!

Several people have helped make Objectivity a success (and great fun).

I'd especially like to thank the Royal Society Library Team, most notably cult hero Keith Moore and Rupert Baker.

Also at the Royal Society, Tracey Hughes, Lesley Miles and Julie Maxton.

The very cool and curious Jake Chudnow at YouTube/Google who got things happening at the start.

And last but definitely not least, James Hennessy who films and edits the videos. He does an amazing job.

PS: Thanks to these guest stars too (click their names to see them on Objectivity)
Sir Paul Nurse, Michael Stevens (Vsauce), Anson at Nasa, J Willgoose Esquire (Public Service Broadcasting), Lucie Green, Charlie McDonnell, Festival of the Spoken NerdSteve Mould, Helen Arney, Matt Parker, Destin Sandlin (Smarter Every Day), Don Pettit (astronaut), Hannah Fry, Mike Merrifield, and Sir Martyn Poliakoff.

 Shooting Number 100 with Keith and James

Shooting Number 100 with Keith and James

Memorial to Caesar

Caesar was my dad's tracking dog during the Vietnam War.

A statue of Caesar has just been unveiled at a railway station in Sydney, near the site of the former Ingleburn Army Camp where he was based pre-service.

 Dad Peter (in white shirt) and other handlers with the statue

Dad Peter (in white shirt) and other handlers with the statue

A second video about Caesar in Vietnam should be coming soon.

 The sculptor

The sculptor

My dad wrote a great book about his experiences with Caesar during the war. It is called Trackers and well worth a read if you like either military stories or just great dog tales!

 Peter and Caesar

Peter and Caesar

 Caesar in Vietnam

Caesar in Vietnam

See also Caesar drawing and dog tag at the Australian War Memorial.

Tragic Deaths

By coincidence we've had a lot of unfortunate deaths on Objectivity in recent weeks.

Most recently, chemist John Daniell who had a stroke at a Royal Society Council meeting. 

His colleagues made the bloody decision to open his jugular there in the meeting room. It did not help.

We also told the story of esteemed scientist John Tyndall who met his demise at the hands of his own wife in a bedside mishap.

And finally Francis Vernon - an intrepid traveller who saw amazing things but came unstuck in an argument over a pen knife.

If you enjoy tales like these - and cool bits of science treasure - do consider watching more from Objectivity.

Mondrian Art Puzzle

The latest Numberphile video concerns a math puzzle based on the artworks for Piet Mondrian (Scroll to bottom of post for some corrections and updates)

Here's the video featuring Gordon Hamilton:

More can be found at Gord's Math Pickle website.

A man who has done some additional work in this area is Ed Pegg Jr.

I had some useful correspondence with him. This demonstration created by Ed is very useful.

And some of our correspondence is shared below with his permission.

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I'm using two methods.  The numerical one can start with the areas of all rectangles that can be cut from a square.  For example, the 9x9  can have subrectangles of area 1, 2, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 9, 10, 12, 12, 14, 15, 16, 16, 18, 18, 20, 21, 24, 24, 25, 27, 28, 30, 32, 35, 36, 36, 40, 42, 45, 48, 49, 54, 56, 63, 64, 72, 81.   One easy cut is 4x5 and 5x5, for a difference of 9.  So look for all ranges less than that, and see if they can be put back together.  
The other method uses graphs.  The number of graphs gets high fast.
http://www.numericana.com/data/polycount.htm  

For example, 1000 by 1000 might need 35 rectangles, with 5986979643542 graphs to check through.  But it might be more. 

Both methods become computationally difficult pretty quickly.  I'm not sure which of my solutions up to 32 can be considered unbeatable. I was just trying for a hard-to-beat baseline.  It probably wouldn't be too hard to do a numerical analysis of the existing solutions and check to see if there was a smaller range that might yield a solution.  

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I should start by mentioning the Mrs. Perkin's Quilt problem.  For a given square, divide it into smaller squares so that the sizes are relatively prime. Solved for smaller values back in the 1950's.  I use the old programs on new computers and extended the results.  Details of the older programs at squaring.net.


http://demonstrations.wolfram.com/MrsPerkinssQuilts/  
From a crushed version of those squares, I developed the Mondrian puzzles  
http://demonstrations.wolfram.com/MondrianPuzzles/  
In PickleMondrain.nb, I give all the solutions I had up to 32.  
In mondrian.xmpuzzle, I give a set-up file for Burr Tools.
http://burrtools.sourceforge.net/    
Let's look at the 6x6.  The range of rectangles 4, 4, 5, 6, 8, 9  might be a solution.  Over in Burr Tools, you can use the 6x6 as a goal and then pick out the rectangles with those areas.  Note that you have two choices for the area 6 rectangle and can only choose one at a time.  Sometimes you might have 40 or more combinations to go though.  the 6x6 is solved readily.  
For the 9x9, there is a known solution with defect 6.  There is a possible split with areas 14, 15, 16, 18, 18.  With Burr Tools, we can prove that split is impossible.  So the 9x9 is proven optimal.  
For 10x10 to 17x17, I pretty much did that.  I may have missed a few combos somewhere, but probably not.  
For the 18x18, there is a defect 10 solution.  There are four area sets that might give a defect 8 solution.  I used Burr Tools to check all the combos, and none of them gave solutions.  So the 18x18 is proven optimal.  
For the 19x19 a defect 11 solution was known.  Just now, I looked at the promising defect 9 case, and found a solution with Burr Tools.   All of the solutions there had two rectangles sharing a full edge, so they wouldn't be found in a search of 3-connected graphs with the electrical method. 
For 20-32, it looks like there are many smaller cases I didn't check.  It would likely be possible to check through all of them by hand within a few days.  My gut feeling is that there might be 2 or 3 improvements if all cases and combos were checked.  

After doing these for a while, you start to appreciate complete sets of rectangles, because that usually means a low defect and only 1 combo to check.  For the 36x36, I noticed that the 24 rectangles of area 48 to 60 might work.  Burr Tools immediately found hundreds of solutions.  

Then I branched out into 3D.  Is it possible to make a polyhedron out of different rectangles?  I managed to do it with 30 rectangles.  

http://community.wolfram.com/groups/-/m/t/928487  
Is it possible to divide a square into different rectangles so that all the diagonal have the same length?  I solved that one, too.  
http://math.stackexchange.com/questions/1958738/same-diagonal-dissection  
Hope that helps.  
POST SCRIPT: After the video was published, Ed improved his 25x25 score from 12 to 11... Here is the result...
POST SCRIPT 2: Ed reports Bruce Norskog found an improvement to n(18).  Ed himself went through and rechecked everything, and also found improvements for n(15) and n(19).

OEIS Sequence: https://oeis.org/A276523