Month: January 2004

Grand Challenges in Mathematics.

In his famous 1900 speech The Problems of Mathematics David Hilbert listed 23 problems that set the stage for 20th century mathematics.

It was a speech full of optimism for mathematics in the coming century and Hilbert felt open (or unsolved) problems were a sign of vitality:

“The great importance of definite problems for the progress of mathematical science in general … is undeniable … [for] as long as a branch of knowledge supplies a surplus of such problems, it maintains its vitality … every mathematician certainly shares … the conviction that every mathematical problem is necessarily capable of strict resolution … we hear within ourselves the constant cry: There is the problem, seek the solution. You can find it through pure thought …”

Hilbert’s problems included the continuum hypothesis, the “well-ordering” of the reals, Goldbach’s conjecture, the transcendence of powers of algebraic numbers, the Riemann hypothesis, the extension of Dirichlet’s principle and many more.

Many were solved in subsequent decades, and each time it was a major event for mathematics.

The Riemann hypothesis (which deals with the distribution of prime numbers) remains on a list of seven “third millennium” problems.

For the solution of each millennium problem, the Clay Mathematics Institute offers – in the spirit of the times – a one-million-dollar prize.

This prize has already been awarded and refused by Perelman for resolving the Poincaré conjecture. The solution also merited Science’s Breakthrough of the Year, the first-time mathematics had been so honoured.

Certainly, given unlimited moolah, learned groups could be gathered to attack each problem and assisted in various material ways. But targeted research in mathematics has even less history of success than in the other sciences … which is saying something.

Doron Zeilberger famously said that the Riemann hypothesis is the only piece of mathematics whose proof (i.e. certainty of knowledge) merits $10 billion being spent.

“In 1965 the Russian mathematician Alexander Konrod said ‘Chess is the Drosophila [a type of fruit fly] of artificial intelligence.

“But computer chess has developed as genetics might have if the geneticists had concentrated their efforts, starting in 1910, on breeding racing Drosophila. We would have some science, but mainly we would have very fast fruit flies.”

Unfortunately, the so-called “curse of exponentiality” – whereby the more difficult a problem becomes, the challenge of solving it increases exponentially – pervades all computing, and especially mathematics.

As a result, many problems – such as Ramsey’s Theorem – will likely be impossible to solve by computer brute force, regardless of advances in technology.

Money for nothing

But, of course, I must get to the point. You’re offering me a blank cheque, so what would I do? A holiday in Greece for two? No, not this time. Here’s my manifesto:

Google has transformed mathematical life (as it has with all aspects of life) but is not very good at answering mathematical questions – even if one knows precisely the question to ask and it involves no symbols.

In February, IBM’s Watson computer walloped the best human Jeopardy players in one of the most impressive displays of natural language competence by a machine.

I would pour money into developing an enhanced Watson for mathematics and would acquire the whole corpus of maths for its database.

Maths ages very well and I am certain we would discover a treasure trove. Since it’s hard to tell where maths ends and physics, computer science and other subjects begin, I would be catholic in my acquisitions.

Since I am as rich as Croesus and can buy my way out of trouble, I will not suffer the same court challenges Google Books has faced.

I should also pay to develop a comprehensive computation and publishing system with features that allow one to manipulate mathematics while reading it and which ensures published mathematics is rich and multi-textured, allowing for reading at a variety of levels.

Since I am still in a spending mood, I would endow a mathematical research institute with great collaboration tools for roughly each ten million people on Earth.

Such institutes have greatly enhanced research in the countries that can afford and chose to fund them.

Content with my work, I would then rest.

For more such insights, log into our website https://international-maths-challenge.com

Credit of the article given to Jonathan Borwein (Jon), University of Newcastle

The diagram above (known as the tetractys) shows the first four triangular numbers (1, 3, 6, 10, …). Although there is a simple formula for calculating these numbers directly, t(n) = 1/2(n(n+1)), constructing them by these layered-triangle diagrams helps to show their geometric and recursive properties.

More generally, polygonal numbers arise from counting arrangements of dots in regular polygonal patterns. Larger polygons are built from smaller ones of the same type by adding additional layers of dots, called gnomons. Beginning with a single dot, k-sided polygons are built by adding gnomons consisting of k-2 segments, with each segment of the gnomon having one more dot than the segments of the previous layer. In this way, the nth gnomon consists of segments each n dots long, but with k-3 dots shared by adjoining segments (the corners).

This post describes how you can draw figures that illustrate the polygonal numbers and explore the polygonal numbers in general (triangular, square, pentagonal, hexagonal, etc.) using either TinkerPlots or Fathom. Both TinkerPlots and Fathom work well, but TinkerPlots creates nicer pictures, and allows for captions directly on the graph.
Without describing the details of how you create Fathom or TinkerPlot documents, here are the attributes that you will want to define in order to draw diagrams like the ones shown.

Required attributes
Create a slider k. This will allow you to set what kind of polygonal number you want to draw (k=3 gives triangular numbers, k=4 gives square numbers, etc.)
Define the following attributes:

n The number itself. This is a natural number beginning at 1 and continuing through the number of cases.

gnomon This states which “layer” or gnomon the number belongs in. It is calculated based on a number of other attributes.

g_index This is the position of the number within the gnonom – it ranges from 1 up until the next k-polygonal number is hit.

s_index Each gonom is broken up into sections or sides – what is the position within the side? Each side is of length equal to the gonom number. The first gonom has sides of length 1, the second has length 2, etc.

corner This keeps track of whether or not the number is a “corner” or not. This is based primarily on the s_index attribute.

c_index This keeps track of how many corners we have so far. There are only k-1 corners in a gnomon (the first number n=1 is the remaining corner). So, when we hit the last corner, we know we are at a polygonal number.

k_poly Records whether or not the number n is k-polygonal. It does this by checking to see if it is the last corner of a gnonom.
The attributes listed above are required for finding the position of each number within the figure; th following attributes are used in actually drawing the figures.

angle The base corner angle for the polygon is determined by k. This is the external angle for each corner.

current_angle We have to add to the base angle at each corner as we turn at each corner. This attribute is used to keep track of the total current angle.

dx This is the x-component of the unit direction vector that we are travelling in. Each new dot moves one dx over in the x-direction. It is given by the cosine of the current angle.

dy This is the y-component of the unit direction vector that we are travelling in. Each new dot moves one dy over in the y-direction. It is given by the sine of the current angle.

prev_g_1_x This is the x-coordinate of the first dot in the previous gonom layer. We need to know this because it will be the starting point for the next layer – each layer starts back at the “beginning” of the figure.

prev_g_1_y This is the y- coordinate of the first dot in the previous gonom layer.

x This is the x-coordinate of the current dot, calculated either from the previous dot or from the first dot in the previous layer.

y This is the y-coordinate of the current dot, calculated either from the previous dot or from the first dot in the previous layer.

caption Used to display the number on the plot (TinkerPlots only)
Below are the formulas for each attribute, written in “ascii” math. They are presented without a full explanation, in the hopes that if you try to implement this you will think about and explore each using the formulas and the descriptions above as a guide. Alternate methods for drawing the diagrams are possible, and you might find other formulas that achieve the same goals. Note that there are nested if() statements in several formulas.

n = caseIndex

gnomon = if(n=1){1, if(prev(k_poly)){prev(gnomon)+1, prev(gnomon)

g_index = if(n=1){ 1, if(prev(k_poly){1, prev(g_index) +1

s_index = if(n=1){ 1, if(prev(k_poly){1, if(prev(s_index) = gnomon){2,prev(s_index)

corner = (s_index=1) or (s_index=gnomon)

c_index= if(g_index=1){1, if(corner){prev(c_index)+1, prev(c_index)

k_poly = if(n=1){true, (c_index=k-1)

prev_g_1_x = if (n=1){0, if(g_index=2){prev(x), prev(prev_g_1_x)

prev_g_1_y = if(n=1){0, if(g_index=2){prev(y), prev(prev_g_1_y)

angle = pi-((k-2)*(pi/k))

current_angle = if(g_index =1) {pi-angle, if(pref(corner)){prev(current_angle)-angle, prev(current_angle)

dx = cos(current_angle)

dy = sin(current_angle)

x= if(n=1){0, if(g_index=1){prev_g_1_x +dx, prev(x) +dx

y =if(n=1){0, if(g_index=1){prev_g_1_y +dy, prev(y) +dy

caption = if(k_poly){n,””

To actually draw the diagrams, create a new plot with the x and y attributes as the horizontal and vertical axis, respectively. Add cases to the collection to populate the diagram. Optionally you can show connecting lines, and (in TinkerPlots) add a legend using the caption attribute.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to dan.mackinnon*