Tomography is the imaging of the interior of an object from information obtained outside it. A familiar example is a CT scan in which a slice of the brain or other organ is imaged from X-ray scans of it. It is not a simple X-ray of the brain but requires mathematics and non-trivial algorithms to produce an image. To explain how it works, try to solve the following problem in the grid below: Given the sums of the rows, columns and diagonals, figure out what the internal numbers are. All the numbers are 1s or 0s. Two of them are shown to start you off.

Clues:

Now enlarge this grid to a 1000 by 1000 and give more clues in the form of more sums of diagonals in all directions; solving this larger problem is precisely what is done in a CT scan.

A CT scan of the brain, image modified from Wikipedia article ‘CT scan’.

The sums are deduced from the X-rays penetrating the body and how much they are attenuated; when the X-ray emitter and receiver rotate around the patient, they are effectively taking sums in all possible directions.

Tomography is a special case of a subject called Inverse Problems, in which one numerically calculates the ‘cause’ from the ‘effect’. It has a long history, starting with a problem posed by Niels Abel in 1826: Can one deduce the profile of a hill by rolling a ball up it at various speeds and noting the return times?

Today this subject pervades modern science because it allows one to explore the insides of an object without destroying it. X-ray tomography, of which CT scans are an example, have revolutionised archaeology and fossil analysis.

Impedance tomography take output currents from input voltages to reconstruct the interior resistance density of an object.

Seismograph tomography can reconstruct the interior density underneath the earth’s surface by measuring the output vibrations after test detonations at the surface, helping in the search for oil, gas or minerals.

The list is long and increasing. Research work done at the University of Malta about deducing internal brain activity from EEG data and published in the Journal of NeuroEngineering and Rehabilitation, Review on solving the inverse problem in EEG source analysis by Roberta Grech et al., has become a standard reference in the field.

Number connections

• Add consecutive odd numbers, starting from 1: you get square numbers, e.g. 1 + 3 + 5 = 9.

• Add consecutive numbers, starting from 1: you get what are called triangular numbers, 1, 1 + 2 = 3, 1 + 2 + 3 = 6, 1 + 2 + 3 + 4 = 10, etc.

• Add two consecutive triangular numbers: you get square numbers again, e.g. 6 + 10 = 16.

• Subtract a number from its square: you get twice a triangular number, e.g. 16 - 4 = 12 = 2 x 6.

• You also get twice a triangular number when you add consecutive even numbers, or multiply two consecutive numbers, e.g. 2 + 4 + 6 = 12, 4 x 5 = 20 = 2 x 10.

• Add consecutive cubic numbers: you get a square, e.g. 13 + 23 + 33 = 36; and it is always the square of a triangular number.

• Multiply two consecutive odd numbers: you always get one less than a square, e.g. 7 x 9 = 63.

For more trivia see: www.um.edu.mt/think

Sound bites

• We have all seen a pyramid-like stack of apples or oranges. The idealised version of this packing is the densest possible for identical spheres in three dimensions. Surprisingly, this fact was only proved in 1998 by T.C. Hales, and even then it took 16 years to verify the 250-page proof completely. Finding the densest packing of ‘spheres’ in two dimensions is not hard (can you think of it?), but in higher dimensions it has been a hard problem for many years. In a major breakthrough last year, Maryna Viazovska solved this problem for 8 and 24 dimensions. More details: www.quantamagazine.org/20160330-sphere-packing-solved-in-higher-dimensions. Another proof that is proving hard to verify is that of the ABC conjecture; it has already been four years since it was published. Is this the future for hard theorems?

• Here are some easy-to-state problems that no one has yet been able to show are true:

Goldbach conjecture: Every even number, except 2, is the sum of two prime numbers.

There are no odd numbers which are the sum of their proper factors (e.g. 6 = 1 + 2 + 3 is the sum of its factors but it is even).

There are an infinite number of prime numbers p such that p + 2 is also prime.

Gilbreath’s conjecture: Start with a list of the prime numbers 2, 3, 5, 7, 11 ...and take their differences: 1, 2, 2, 4, 2 ...; and repeat 1, 0, 2, 2 ..., and again 1, 2, 0, 0 ....The initial number of these sequences is conjectured to be always 1.

• For more science news, listen to Radio Mocha on Radju Malta 2 every Monday at 1pm and every Friday at 6pm.