Purging away from appalling reminiscences,

Hauling out her soul from such sticky black-hole,

She needed someone to hold.

Picking, stitching her teary, weary capacitances,

An unsteady lone particle, travelling hefty distances,

Tearing herself away from her soul- like sticky charcoal,

She needed someone to hold.

And be told that she was worth.

Worthy of all the love in the world and in his heart,

Of all the pressure she carried of affection that tore her apart,

Of the warmth and loyalty till the end from the start,

That she was beautiful like mathematics, like an art.

Struggling to brace her ground on whizzing, rotating surrounding,

Losing vigor of her keens in a world insanely astounding,

Vanishing in the night darker than her kohl,

She needed someone to hold.

Kicking, pricking her ribcage becoming uninhabitable compounding.

Emerging dust and fume making her heart deeply pounding,

Her sturdy neural flux quickly turning bipole,

She needed someone to hold.

And be told that she was worth.

Worthy of all the love in the world and in his heart,

Of all the pressure she carried of affection that tore her apart,

Of the warmth and loyalty till the end from the start,

That she was beautiful like mathematics, like an art.

She was beautiful; she often tended to forget.

She didn’t need anyone to tell her that.

She was beautiful just like mathematics,

Just unsolved yet.

To me, what was interesting about it was its simplicity. You start by defining a formal system, i.e. variables, symbols, axioms, inference rules. Then you build new theorems based on the formal system.

The core concept behind Metamath is substitution, and it’s using RPN notation to build hypothesis on the stack and then rewrite them using the inference rules to reach conclusion.

The program is written in C, and I compiled it on both OS X and my Android phone. It’s pretty light-weight and compiles in a few milliseconds, and it’s also interesting that you can take it anywhere you want on your phone.

Metamath has no special syntax. It will tokenize a file that we give to it, and tokens that start with `$`

are Metamath tokens, while everything else are user-defined tokens.

Here is a list of built-in Metamath tokens:

- To define constants, use the
`$c`

token. - To define variables, use the
`$v`

token. - To define types of variables, use the
`$f`

token. - To define essential hypothesis, use the
`$e`

token. - To define axioms, use the
`$a`

token. - To define proofs, use the
`$p`

token. - To start proving in
`$p`

statement, use the`$=`

token. - To end the statements above, use the
`$.`

token. - To insert comments, use the
`$(`

and`$)`

tokens. - To define a block (has effect on scoping), use the
`${`

and`$}`

tokens. Note that only`$a`

and`$p`

tokens will remain outside the scope.

That’s basically it. The package has included demo example and an example for the MU puzzle as well. There are other systems as well (Peano, Set).

Now for the example we’ll use, start by creating a test.mm file. We’ll define a formal system and demonstrate the usage of modus ponens to come up with a new theorem, based on our initial axioms.

$( Declare the constant symbols we will use $) $c -> ( ) wff |- I J $. $( Declare the variables we will use $) $v p q $. $( Specify properties of the variables, i.e. they are wff formulas $) wp $f wff p $. wq $f wff q $. $( Define "mp", for the modus ponens inference rule $) ${ mp1 $e |- p $. mp2 $e |- ( p -> q ) $. mp $a |- q $. $} $( Define our initial axioms. I and J are well-formed formulas, we have a proof for I and we have conditional for I -> J $) wI $a wff I $. wJ $a wff J $. in1 $a |- I $. in2 $a |- ( I -> J ) $. $( Prove that we can deduce I from the initial axioms $) proof_I $p |- I $= in1 $. $( Prove that we can deduce J from the initial axioms $) proof_J $p |- J $= wI $( Stack: [ 'wff I' ] $) wJ $( Stack: [ 'wff I', 'wff J' ] $) $( Note that we had to specify wff for I and J before using mp $) in1 $( Stack: [ 'wff I', 'wff J', '|- I' ] $) in2 $( Stack: [ 'wff I', 'wff J', '|- I', '|- ( I -> J )' ] $) mp $( Stack: [ '|- J' ] $) $.

To verify our file, we run `./metamath 'read test.mm' 'verify proof *' exit`

.

Note how we had to separate `wff`

from `|-`

. Otherwise, if we just used `|-`

, then all well formed formulas would be proven to be true, which doesn’t make much sense :)

For a more complete example, you can check out my peano.mm where I define successor for natural numbers and prove some theorems about ordering of them.

Due to its minimalistic design, when compared to Coq it has no Calculus of Constructions, Inductive Types, etc.

To conclude, I think it’s a fun program to play with, but since it has no “real” framework, I don’t think it’s as industry ready as Coq.

]]>So I’m married to a mathematician, and have spent the past 10 or so years hearing all about the great numbersmiths of yore and nodding along politely. I thought I knew everything I needed to know about, for example, Leonhard Euler (specifically, that his name is pronounced ‘oiler’).

But then I came across this picture.

Which prompted the following exchange with my husband.

He then shared a series of pictures of questionably-hatted Great Mathematicians. And that’s how I learned that Mike’s been holding out on me this whole time. It turns out that with one exception, all mathematicians ever have sucked at wearing things on their heads.

Here’s Gauss, famously the last person to know all of math as it was known at the time. Unfortunately, I gather, he wasn’t sure what to do with the bag his whisky came in. (Many of us can relate.)

Here’s Fibonacci, who introduced the strangely beautiful Fibonacci sequence to Western audiences. His affinity for order & beauty did not extend to his headgear, which seems to have been an early inspiration for Euler’s boxers.

There’s an entire sub-genre of mathematicians who forgot to take off their swim caps before heading back to the office.

Unsurprisingly, Leibniz (L) > Newton (R) in every respect including wig fabulousness. Neither, however, could top their spherical Scottish contemporary, Colin Maclaurin. I’d suggest that this crushing failure could explain a lot about Newton’s famous eccentricity.

It would appear that David Hilbert picked up on the efforts of his many forebears and weighed in with this respectable contribution.

But only one mathematician perfected the art of cranial adornment. Just look at that ribbon, those jaunty flowers, those exquisite Princess Leia buns. There’s even a veil cascading from this masterpiece, daring the eye to drift away from the top of Our Queen’s brilliant head. This is Ada Lovelace: mathematician, writer, only legitimate child of Lord Byron, first computer programmer, and goddess. She puts Leonhard Euler’s head-boxers to shame.

Maybe someday I’ll forgive Mike for keeping all this from me for so long. It appears to have been an honest, if egregious, mistake. And if there’s a mathematician in your life who’s been likewise shielding you from The Hats, I’m sorry you had to find out this way.

]]>TOTP is used primarily with Google Authenticator mobile app. But the algorithm can be easily implemented. **All that is needed is the key** provided by the internet resource we want to access to. Only using that alphanumeric key and the current time, a six digit value is calculated which acts as a second step verification for access granting. This way, a password snooping, even if it involves the TOTP value, cannot be reused: that’s because the **TOTP changes every 30 seconds** and cannot be predicted from previous values – it can only be calculated if the key is known, and the key itself is only interchanged when the 2-step verification is first activated.

The HOTP algorithm involves the use of SHA-1.

An interesting particularity here is the use of **Unix time** and the “problem” of the leap seconds… So as an oddity, a particular TOTP token used at the end of June or December in next years could be valid for 31 seconds. More exactly, if a leap second is added, the Unix time for the first second of the next day will be repeated twice, so that minute will really have two 0 seconds…

Anyway, as I wanted to obtain TOTP tokens via command line, but didn’t want the key to be directly visible in the code, I made a python script which can be configured to use mangled TOTP keys in code: the keys are “encrypted” with XOR and a random ASCII key.

This way a casual inspection of the code or an automatic hacking won’t directly obtain the key… Of course this can be discussed :-)

**The script twisted2sv.py can be obtained at github,** and requires its rewriting to first insert the keys and choose a XOR key if desired (the variable

Once configured, and after a first run to autorewrite itself with encrypted keys, **it will print on screen TOTP tokens for all your sites**, indicating the time in seconds that they will last… after each 30 s interval new values will be printed in screen. By default two consecutive 30 s intervals are printed: this should allow easy use of the tokens for accessing any site.

]]>$ vi twisted2sv.py#<-- insert your TOTP key(s)$ python3 twisted2sv.pyAuto-phagocytizing to encrypt TOTP keys ... Done.$ python3 twisted2sv.py 4#<-- print 4 sets of tokens (4*30 = 2 minutes time)site1: (1) 973722 my site 2: (1) 008862 site1: (2) 862833 my site 2: (2) 274628(13) <-- this decremental counter tells you the seconds until these tokens' death

What is Supreme Mathematics, and why should we use it? ALLAH’s Supreme Mathematics is the numbers 1-9 plus 0 and the true definitions that go along with each one of them: **(1)KNOWLEDGE (2)WISDOM (3)UNDERSTANDING (4)CULTURE/FREEDOM (5)POWER/REFINEMENT (6)EQUALITY (7)GOD (8)BUILD/DESTROY (9)BORN (0) CIPHER. **Using mathematics teaches us that we must add up things correctly in order to get a right answer IN ALL THINGS. For instance when you add Knowledge on to Knowledge you gain Wisdom in the process (1+1=2). Wisdom is the application of knowledge. The use of mathematics daily gives you the chance to define the principles of the numbers each day holds i.e. the date. Here’s an example, today’s date is the 16th, so today’s Math is **(1)KNOWLEDGE (6)EQUALITY = (7)GOD. **As you gain more knowledge, wisdom, an understanding of these definitions, the more your insight of these principles will grow. Showing and proving mathematics to be right and exact.

I will build on a few examples to prove my point. The number 1 can also be represented as the Man, the number 2 can be represented as the Woman, and the number 3 is the Child. Adding a (1)Man plus a (2) Woman gives you a (3)Child (1+2+3). Also adding (2)**WISDOM** to your **(1)KNOWLEDGE **gives you **(3)UNDERSTANDING **(2+1=3). Here’s an example that shows how Mathematics is even in correlation to the universe: we are 93 million miles away from the Sun, (9)Born (3) Understanding. If you add 9+3 it will give you 12 (which is not a base number, so we must add the two base numbers in order to get the true understanding of the original problem). So 1+2 gives you 3. Therefore (9) BORN plus (3)UNDERSTANDING equals (1)KNOWLEDGE (2)WISDOM which equals (3)UNDERSTANDING. In other words the distance of our planet from the Sun represents UNDERSTANDING. The Earth is also the third planet away from the Sun, and that means our placement in our solar system represents UNDERSTANDING. By applying mathematics to this distance we are able to view our life from a different aspect. Our planet is the only one that host intelligent life in the universe, the Earth is in the perfect position in relevance to our Sun to produce an atmosphere that can support life as we know it. The Sun and Earth have been in this position billions of years before Man existed . If Supreme Mathematics were not right and exact how could these observations make any sense. Adding (1)Man to (1)Man will not give you a (3)Child, neither will adding (2)Woman to (2)Woman, only when adding (1)Man to (2)Woman will to get (3)Child. As you begin to master the use mathematics, you begin to master self.

P.E.A.C.E.

Knowledge Supreme Intelligence

]]>**History**

Kaliningrad is a small town in Russia, which was known as Konigsberg in eighteenth century or earlier. The town was built by the flow of the river Pregel. A big island was at the middle of the river. It was divided into two streams due to the island, got united again and continued in two directions again. Habitats grew on both sides and on the island. There were many bridges built to connect people from both sides for life. Actually seven bridges were built. The island was called Kniphof. The seven Konigsberg bridges were used to connect three land parts by two sides of the river and Kniphof. On a holiday, a long walk could cover all these bridges one after another. The dwellers in the town were happy enough to walk through the bridges and feel the nature in those days. Gradually a maze and ultimately a challenge had been developed about crossing all of the bridges starting from one point and coming back to the same. Can a person start walking from a point and come back to the same by crossing all seven bridges only once? Nobody was successful though many tried years after years in early eighteenth century. Al last in 1736, the famous Swiss Mathematician Leonard Euler made an explanation for this puzzle by a mathematical theory.

**Graph Theory**

The Mathematics behind this problem of Konigsberg Bridges is Graph Theory. Mathematicians have worked and been working on this for solving many real life problems so far. Sometimes the problems are so complicated that computer programs are necessary to process calculations. Computer programs were not used in eighteenth century when Leonard Euler (1707-1783) first wrote a scholarly article on Graph Theory, which was not then a recognized branch in Mathematics. A graph is actually a system of points with some specific connections. Graph is not same as chart sometimes people get confused with. Graph expresses relationships amongst points, such as, computer network at an office or road communication among cities of a state. Here each of the cities or points is called a vertex and each connecting road or line is called an edge. In a graph, every point may not have equal number of edges. A graph can be written as a set like {4, 3, 2, 2, 1}. Here there are five points in the graph, number of edges associated with each point are shown in the set in descending order. Sum of the numbers, that is, sum of the degrees, in such set is always even and double of the number of edges in a graph. A sequence of vertices along with their edges is called a path. A circuit can be made when a path starts from a point and ends at the same. Eulerian Path of a graph crosses all the edges of a graph by reaching vertices one after another, but no edge is crossed more than once. Eulerian Circuit is the Eulerian Path, which starts from and ends at a same vertex.

**Solution**

Eulerian Circuit may exist for a graph if and only if each vertex is of even degrees. {4, 2, 2, 2, 2} is an example Eulerian graph, where you can start from a point and come back to the same by crossing all the edges only once. Eulerian Path may exist for a graph when there are only two vertices with odd degrees. These are called semi-Eulerian graph. {4, 3, 2, 2, 1} is an example of semi-Eulerian graph, where you can start from an odd degree vertex, 3 or 1 in this case, and reach at the other by crossing all the edges only once.

Our Konigsberg Bridge problem is graph with four vertices as the four land parts. Each land part is connected to another through bridges. So the graph is represented as a sequence all odd numbered vertices as {5, 3, 3, 3}. This is neither an Eulerian nor a semi-Eulerian graph for which Eulerian Circuit or Eulerian Path exists. That means, it is not possible to start from a land part and come back to the same by crossing all the seven bridges only once.

]]>**Graph Theory**

The Mathematics behind this problem of drawing in one go is Graph Theory. Mathematicians have worked and been working on this for solving many real life problems so far. The problems may be simple and small like this kids’ drawing puzzle. Sometimes the problems are so complicated that computer programs are necessary to process its huge volume of calculations. The Graph Theory is credited to a Swiss Mathematician Leonard Euler (1707-1783) who first wrote a scholarly article on Graph Theory relating to a bridge problem called Konigsberg Bridge Puzzle. Later this was well accepted as a significant branch in Mathematics. A graph is actually a system of points with some specific connections. Graph is not same as chart sometimes people get confused with. Graph expresses relationships amongst points, such as, computer network at an office or road communication among cities of a state. Here each of the cities or points is called a vertex and each connecting road or line is called an edge. In a graph, every point may not have equal number of edges. A sequence of vertices along with their edges is called a path. A circuit can be made when a path starts from a point and ends at the same. Eulerian Path of a graph crosses all the edges of a graph by reaching vertices one after another, but no edge is crossed more than once. Eulerian Circuit is the Eulerian Path, which starts from and ends at a same vertex.

**Solution**

To draw a figure (graph) in one go, there must exist a Eulerian Circuit for the graph. It is possible if and only if each vertex of that graph has even degrees. In the given figure, the flower graph can be presented as {8, 2, 2, 2, 2, 2, 2, 2, 2, 2} in which each and every vertex has even number of degrees. So one can start from any point and move through edges and come back at the starting point in one go. This is Eulerian graph because there exists the Eulerain Circuit. Similarly, a graph can be called semi-Eulerian if there exists only a Eulerian Path. Eulerian Path may exist in a graph when there are only two vertices with odd degrees. The hat {3, 3, 2, 2, 2, 2} is an example of semi-Eulerian graph, where one can start from an odd degree vertex, A or B in this case, and reach at the other by crossing all the edges only once; without lifting the pencil up. In our given figure, the house is a {3, 3, 3, 3, 2, 2, 2} graph in which there are more than two odd degree vertices. As a result, this house can not be drawn in one go because there does not exist Eulerian Circuit or Eulerian Path in this graph according to the Graph Theory.

]]>It’s a ‘Platonic solid in 4 dimensions’ with 600 tetrahedral faces and 120 vertices. One reason I like it is that you can think of these vertices as forming a *group*: a double cover of the rotational symmetry group of the icosahedron. Another reason is that it’s a halfway house between the icosahedron and the lattice. I explained all this in my last post here:

I wrote that post as a spinoff of an article I was writing for the *Newsletter of the London Mathematical Society*, which had a deadline attached to it. Now I should be writing something else, for another deadline. But somehow deadlines strongly demotivate me—they make me want to do *anything else*. So I’ve been continuing to think about the 600-cell. I posed some puzzles about it in the comments to my last post, and they led me to some interesting thoughts, which I feel like explaining. But they’re not quite solidified, so right now I just want to give a fairly concrete picture of the 600-cell, or at least its vertices.

This will be a much less demanding post than the last one—and correspondingly less rewarding. Remember the basic idea:

Points in the 3-sphere can be seen as quaternions of norm 1, and these form a group that double covers The vertices of the 600-cell are the points of a subgroup that double covers the rotational symmetry group of the icosahedron. This group is the famous **binary icosahedral group**.

Thus, we can name the vertices of the 600-cell by rotations of the icosahedron—as long as we remember to distinguish between a rotation by and a rotation by Let’s do it!

• 0° (1 of these). We can take the identity rotation as our chosen ‘favorite’ vertex of the 600-cell.

• 72° (12 of these). The nearest neighbors of our chosen vertex correspond to the rotations by the smallest angles that are symmetries of the icosahedron; these correspond to taking any of its 12 vertices and giving it a 1/5 turn clockwise.

• 120° (20 of these). The next nearest neighbors correspond to taking one of the 20 faces of the icosahedron and giving it a 1/3 turn clockwise.

• 144° (12 of these). These correspond to taking one of the vertices of the icosahedron and giving it a 2/5 turn clockwise.

• 180° (30 of these). These correspond to taking one of the edges and giving it a 1/2 turn clockwise. (Note that since we’re working in the double cover rather than giving one edge a half turn clockwise counts as different than giving the opposite edge a half turn clockwise.)

• 216° (12 of these). These correspond to taking one of the vertices of the icosahedron and giving it a 3/5 turn clockwise. (Again, this counts as different than rotating the opposite vertex by a 2/5 turn clockwise.)

• 240° (20 of these). These correspond to taking one of the faces of the icosahedron and giving it a 2/3 turn clockwise. (Again, this counts as different than rotating the opposite vertex by a 1/3 turn clockwise.)

• 288° (12 of these). These correspond to taking any of the vertices and giving it a 4/5 turn clockwise.

• 360° (1 of these). This corresponds to a full turn in any direction.

Let’s check:

Good! We need a total of 120 vertices.

This calculation also shows that if we move a hyperplane through the 3-sphere, which hits our favorite vertex the moment it touches the 3-sphere, it will give the following slices of the 600-cell:

• Slice 1: a point (our favorite vertex),

• Slice 2: a dodecahedron (its 12 nearest neighbors),

• Slice 3: an icosahedron (the 20 next-nearest neighbors),

• Slice 4: a dodecahedron (the 12 third-nearest neighbors),

• Slice 5: an icosidodecahedron (the 30 fourth-nearest neighbors),

• Slice 6: a dodecahedron (the 12 fifth-nearest neighbors),

• Slice 7: an icosahedron (the 20 sixth-nearest neighbors),

• Slice 8: a dodecahedron (the 12 seventh-nearest neighbors),

• Slice 9: a point (the vertex opposite our favorite).

Here’s a picture drawn by J. Gregory Moxness, illustrating this:

Note that there are 9 slices. Each corresponds to a different conjugacy class in the group These in turn correspond to the dots in the *extended* Dynkin diagram of which has the usual 8 dots and one more.

The usual Dynkin diagram has ‘legs’ of lengths and

The three legs correspond to conjugacy classes in that map to rotational symmetries of an icosahedron that preserve a vertex (5 conjugacy classes), an edge (2 conjugacy classes), and a (3 conjugacy classes)… not counting the element That last element gives the extra dot in the *extended* Dynkin diagram.

One of the most important points of variation that concerns academic life is the school system students go through before going to University. In the system operating in England and Wales the standard qualification for entry is the GCE A-level. Most students take A-levels in three subjects, which gives them a relatively narrow focus although the range of subjects to choose from is rather large. In Ireland the standard qualification is the Leaving Certificate, which comprises a minimum of six subjects, giving students a broader range of knowledge at the sacrifice (perhaps) of a certain amount of depth; it has been decreed for entry into this system that an Irish Leaving Certificate counts as about 2/3 of an A-level for admissions purposes, so Irish students do the equivalent of at least four A-levels, and many do more than this.

There’s a lot to be said for the increased breadth of subjects undertaken for the leaving certificate, but I have no direct experience of teaching first-year university students here yet so I can’t comment on their level of preparedness.

Coincidentally, though, one of the first emails I received this week referred to a consultation about proposed changes to the Leaving Certificate in Applied Mathematics. Not knowing much about the old syllabus, I didn’t feel there was much I could add but I had a look at the new one and was surprised to see a whole `Strand’, on *Mathematical Modelling with netwworks and graphs*.

The introductory blurb reads:

In this strand students learn about networks or graphs as mathematical models which can be used to investigate a wide range of real-world problems. They learn about graphs and adjacency matrices and how useful these are in solving problems. They are given further opportunity to consolidate their understanding that mathematical ideas can be represented in multiple ways. They are introduced to dynamic programming as a quantitative analysis technique used to solve large, complex problems that involve the need to make a sequence of decisions. As they progress in their understanding they will explore and appreciate the use of algorithms in problem solving as well as considering some of the wider issues involved with the use of such techniques.

Among the specific topics listed you will find:

- Minimal Spanning trees applied to problems involving optimising networks and algorithms associated with finding these (Kruskal, Prim);
- Bellman’s Optimality Principal to find the shortest paths in a weighted directed network, and to be able to formulate the process algebraically;

For the record I should say that I’ve actually used Minimal Spanning Trees in a research context (see, e.g., this paper) and have read (and still have) a number of books on graph theory, which I find a truly fascinating subject. It seems to me that the topics all listed above are all interesting and they’re all useful in a range of contexts, but they do seem rather advanced topics to me for a pre-university student and will be unfamiliar to a great many potential teachers of Applied Mathematics too. It may turn out, therefore, that the students will end up getting a very superficial knowledge of this very trendy subject, when they would actually be better off getting a more solid basis in more traditional mathematical methods so I wonder what the reaction will be to this proposal!

]]>

Girls Olympiad – Six girls took part with 3 achieving results within the top 600 in the country (Alex McLain, Kara Salih, Chloe Thorne), and one achieving a result placing her in the top 25 (Amelia Rout).

Senior Maths Challenge – The school entered 100 students, achieving 3 Gold, 16 Silver, 41 Bronze awards, special highlights go to Amelia Rout, Kara Salih, Joshua Cheng all qualifying for round 2 in year 13. Ryan Bushell, Amy Walton both qualifying for round 2 in year 13.

Senior Team Challenge – We took a team of 4 mathematicians (Amelia Rout, Kara Salih, Alex McLain, Amy Walton) to compete against other teams in the region at Ravens Wood School. The team worked very hard and achieved 5^{th} place out of 33 teams, an excellent result.

Congratulations to all Chis and Sid’s highly talented musicians.

]]>

Here’s a silly fragment of a song

(my apologies to bass players, I really LOVE bass & this is non-pejorative!)

Who do you think you are

Rocking hard with guitar

Get out of my face

You should be playing bass…

<3

]]>