What is a triangle? (Or what does the set of triangles "look like")

I recently had a discussion with friends on how much more common obtuse triangles are than acute. It depends of course on how you define a probability measure on triangles, and one has to make some assumptions. For some reasonable choices the answer is obtuse triangle are three times more common than acute. See for example the paper by Guy [1] and the discussion on Stack Exchange[2]. A figure from [1] is revealing. Suppose you break a stick in to three parts. They may or may not form a triangle depending it they satisfy the triangle inequality eg if a> b+c, but when they do they line in the middle triangle of the figure below. The curves are triples a,b,c of side lengths that satisfy Pythagoras' theorem (three hyperbolas) so give right angle triangles. The outside is obtuse and the inside is acute.

We see in this figure that a point represents a triple a,b,c, with a normalization a+b+c=1, and so we have a way to represent a triangle up to translation and similarity. So my question is what is a good way to think of "a triangle" modulo this equivalence relation. In a sense "what is a triangle". Stewart [3] gives a good answer to this. The "moduli space of triangles", and this turns out to be itself a triangle. In a way the "set of triangles is a triangle".

So in this case we consider the shaded area Phi as being the "representative" set of triangles, as others can be obtained by permutation of coordinates. Again the curve corresponds to right angle triangles. Now it is clear from the picture that in a uniform probability measure in these coordinates the obtuse triangles are more common.  

References
[1] Guy, Richard K. "There are three times as many obtuse-angled triangles as there are acute-angled ones." Mathematics Magazine 66.3 (1993): 175-179.
[2] brick (https://math.stackexchange.com/users/187522/brick), Probability of triangle to be acute?, (version: 2015-04-24)
[3] Stewart, Ian. "Why do all triangles form a triangle?." The American Mathematical Monthly 124.1 (2017): 70-73

Three position lines don't meet. Where are you?

I think this might be a fact that is "well know to those who know". I often give an example when teaching inverse problems of three equations in two unknowns, and we  calculate the least squares solution using the Moore-Penrose generalized inverse.

I also give an example from navigation. Suppose you have three non intersection position lines and you draw them on your chart (map). You got these by measuring the angle between stars and the horizon. They don't exactly meet as you have errors in your measurements.

Suppose that the errors in the measurements were normally distributed with the same standard deviation. For example we took enough sextant   readings and averaged and evoked the central limit theorem to say they should be normal.

So which point do we choose in the triangle as our position? The "middle"? The Admiralty Manual of navigation suggests choosing the corner of the triangle closest to danger. But if we are way out in a featureless sea we just want to choose the best.

Well there are 4 well-known centres of triangles (centroid, orthocentre incentre, circumcentre) , and actually there are thousands of interesting centres of triangle with different properties tabulated by Kimberling

The Maximum Likelihood estimate (ie in a sense the most likely) is the one that minimizes the sum of squares of the distances from the position lines.

This point (number 6 in Kimberlings list! ) is called the Symmedian Point, or Lemoine point or the Grebe point

Diagram on Wolfram


Here is "Kimberling's List" (scroll down to "X(6)")

So that is where you are "most likely" to be!

Interestingly it can be constructed fairly easily with ruler and compasses.

One year post doc position on maths of landmines

Thanks to a Royal Society Challenge grant we have a one year post doc to work on the mathematics of metal detection to improve land mine clearance. (Starting as soon as possible - recruitment closes January 2017)

Job advert here

We are concentrating especially on the polarization tensor for the eddy current approximation to Maxwell's equation. See the following papers

• Ammari, H., Chen, J., Chen, Z., Garnier, J. & Volkov, D. (2014) Target detection and characterization from electromagnetic induction data. J. Math. Pures Appl., 101, 54–75 
•  Paul D. Ledger and William R. B. Lionheart
  Characterizing the shape and material properties of hidden targets from magnetic induction data IMA J Appl Math first published online June 26, 2015 doi:10.1093/imamat/hxv015 
•  P. D. Ledger and W. R. B. Lionheart, "Understanding the Magnetic Polarizability Tensor," in IEEE Transactions on Magnetics, vol. 52, no. 5, pp. 1-16, May 2016.doi: 10.1109/TMAG.2015.250716  

We hope to use elements of Timo Betcke's boundary element code BEM++

Netgen on a Mac called from MATLAB

One of the many irritations of MATLAB on a Mac is that calling a command from the system function runs the command in a shell with messed up environment variables. In EIDORS we call Netgen to create meshes.

There are several environment variables I needed to get this to work.
Of course I needed to set the Netgen path for example

 export NETGENDIR=$HOME/netgen/bin 

The equivalent of Linux's LD_LIBRARY_PATH is needed for the Togl library export DYLD_LIBRARY_PATH=$DYLD_LIBRARY_PATH:$HOME/lib/Togl1.7 

Then it still can't find Tik so you need to set

 TCLLIBPATH=$HOME/lib 

You can put that all in s string separated with semi colons and then the netgen command eg

$NETGENDIR/netgen 

and it works.

Open Science

Ideally, where possible, data from scientific experiments, especially published work, should be accessible so that at the very least other researchers can attempt to reproduce your results. I think this is especially true in computational projects, and in that case ideally the source code of the software you used should be accessible.

Another good reason to make experimental data available is that that there are many groups working in scientific computing and applied mathematics without good contacts with experimental researchers, and indeed vice versa. Furthermore widely accessible data sets mean that groups can compare their algorithms and software on the same data set to give a more objective comparison.

CERN's web site Zenodo is an excellent resource for exchange of open data. It is free to use and gives a Digital Object Identifier DOI for the data, making it citable.

In the EIDORS project we have had a data repository for a while now and included some classic data sets including the first EIT (electrical impedance tomography) data of a human, Rod Smallwood's arm. testing the water I transferred the data to Zenodo here.  I think in the future EIDORS contributed data sets might go this way with a link from the EIDORS web site.

My student Sophia Coban recently created a data set of glass beads using the x-ray CT facility in the Henry Mosley Centre here in Manchester. Her data set SophiaBeads is now on Zenodo, with a separate entry for the source code of demo reconstructions. The source code is a single release taken from GitHub, and archived on Zenodo which gives it a DOI.

I am still trying out these open science ideas myself, but I just uploaded my own MRI head scan to Zenodo. Some of my colleagues might think this is too open. Perhaps I want my head examining?  Well why not! However there appears to be nothing especially wrong with my brain, and we will use it as a test for some segmentation and meshing algorithms.

Thumb nail image from MRI of my head.

Why I deleted my research gate account

I was annoyed that Google searches of the titles of some of my papers and others I am interested in provides links to ResearchGate. but there is no content there. Indeed if you log in it invites users to pester the authors to post their content there.

So what they are trying to be is a gate to free content, in exchange for information on your research interests and contacts.

Hopefully this dubious web site will be put out of business in due course by the rise in Open Access scientific publications.

I don't mind putting a preprint on a preprint server like ArXiV or an institutional server provided Google Scholar finds it.   The last thing I want is another site where I need o post identical copies of my preprints and maintain with information, especially more giving that information over to the control of a dubious commercial entity.

...lets hope the constant stream of Research Gate email spam stops as well.

More reading Adam Rechless on his deletion of RG account.

Pros and cons from Exeter Open Ressearch Blog

Discussion on StackExchange

Tell me of any other useful links to discussions...

A new theory for metal detectors?

Following our MIRAN Land Mine clearance meeting in Manchester in 2012 there has been a collective effort to understand how to characterize conducting objects in low frequency magnetic fields. This is the metal detector length scale so without the object the problem is magneto-quasi static and the object satisfies the Eddy Current Approximation.

The search for the right theory began with Tony Peyton and my work on airport metal detectors that can locate and characterize metallic objects. This was in collaboration with Rapiscan and the project was called EMBody. It resulted in  a successful prototype walk through metal detector which was trailed in Manchester Airport (who were also project partners).

So when Sir Bobby Charlton approached us to help with land mine detection, resulting in the founding of the charity Find A Better Way we though first to improve metal detectors so they could locate multiple objects with multiple coils and discriminate between objects based on tehir shape and electrical properties (conductivity and permeability).

This resulted in a long campaign by many mathematicians to understand this case. Habib Ammari at the Ecole Normale Supérieure Paris was at an advantage. With  A. Buffa and J.-C. Nédélec he had written a paper A justification of the eddy currents model for Maxwell's equations.,
which laid the theoretical foundation for the eddy current model. With Kang he had also litterally written the book on Polarization Tensors, a way of representing the response of an object in a field using a small number of coefficients. The problem was to a large extent cracked in Ammari et al's paper Target identification using dictionary matching of generalized polarization tensors.  Interestingly the metal detector, especially Unexploded Ordnance (UXO) community had long argued heuristically that the response to a conductive object should be given by a rank 2 symmetric tensor (six numbers) . Ammari's paper uses a rank 4 tensor, which could be 81 numbers, but with symmetries cuts down to 27. My own work with Paul Ledger from Swansea now shows that this is in fact a rank two tensor in disguise and the UXO literature was right all along. In a way everyone was right! Now we can calculate this tensor from the shape and electromagnetic properties of a material and work towards better metal detectors for land mine clearance.