Pictorial Representation of 7-digit Phone Numbers Thursday, Jan 26 2012 

Disclaimer: I liberally added fonts and images from online. No copyright infringement intended.

I often wondered if it is possible to create glyphs (pictorial representation) of all combination of telephone numbers that can be decoded at glance. Unless the number consists of familiar patterns, form, sequence it becomes frustratingly difficult to commit them to memory driving at 65mph on freeway.

First let us introduce the glyphs for base (0,1,2….9):

1

2

3

4

5

6

7

8

9

Following reveals the whole spectrum of numbers with a specific BASE (in this case 2):

Array of Telephone Numbers in Base 2

Of course the various types of niche or arches represents different form relative to “BASE 2”.  If the same arches appear at the upper-right or left corner (depending upon pattern) then it will denote different numbers per BASE.

These are all very arbitrary; so we can use any general form for any specific arch.  We shall use the special case of Magritte’s man for the sequence which are all of the same digit.

Denotes a sequence of digits that are all same

Let’s us now consider different variations of a 3-digit number.

It can be increasing. Such as 123
Or decreasing: 321
If neither, then we can have a palindrome like 323.
Form of 323 reversed can be 232. Call it reverse or obverse palindrome.
We may have all same 333.
We can also have a double double unit such as 662. Reversing it gives 266.

Enough rambling; let’s play with some random 3-digit numbers. Using some random random generator from internet I get a list whose general form I have verbally described:
206 = Turns out we do not have a convention yet. But a guiding factor is 0 which splits a 26. Now this 26 can be thought of as 13×2. What does 13 do? Well it’s is 1 less of 14, or sub-base minus 1 of 714, my city area code. Note: We will try to think backwards to this CODE of 714. Since we are yearning to be as intuitive as possible, feel free to derive it to a pattern based on your area code.
312 = Again, it has not been defined yet. But to me a pattern that may strike out is that it is a increasing sequence which has been shifted to the right.
422 = This one we have in the bag. It is a unit double double form.
534 = This is an increasing sequence shifted to the right.
653 = It is decreasing sequence.
305 = This has the same form as 206.
844 = Unit double double pattern.
304 = It is 1 minus 305.
852 = Decreasing pattern.
968 = This one is a weird one. We will return to it later.

Okay, so how do we generalize to an algorithm given any sequence of 3-digit number?

Well, let’s see. Right now, word count is 490. This can be thought of as 7 squared zero, giving us 7 for the first digit of my area code 714. But how to verbally pin it down forward?

STEP 1: UNIT(BASE)
STEP 2: SQR(R)
STEP 3: ADDZERO(R)

which roughly translates as take the unit of BASE which I am redefining to be 714. Note, previously we talked of a different base. So, take the unit of BASE and SQUARE the R(esultant), then adding a zero to the end result.

How to represent 094? Simple. Call the first algorithm BUTTERFLY. And then use the function REVERSE.
Or REV(BUTTERFLY).

Things are quickly getting muddy. So let us back-out a second, and get a random 7-digit number. Certain plumber in a certain state has a number 439-3407.

What pattern seems to jump out (bearing well in mind it may vary individually?). It can be grouped to: 4,39,34,07 because the 3-digit sequence has no basic pattern.

Although it is of different state, I will affix it to my area code. Then we will create an algorithm. First , it’s time for some terminologies. If we take the CODE 714 and call it BASE, then there are components called UNIT or 7. DBLUNIT or 14. Singleton or 1. SUBUNIT or 4. SUBBASE or 71.

To unleash an inner Bhaskara, here is the algorithm. Extracting SUBUNIT, subtract 1 to get and putting it at the end, then reverse it around 9 ending it in add-1 of it and sub-2 of it.

Quite a mouthful. To formulate it in steps:

STEP 1: SUBUNIT(BASE)
STEP 2: SUBUNIT(BASE),SUBUNIT(BASE)-1
STEP 3: WRAPAROUND(SQUARE(2,2), {here (2,2) denotes step 2, position 2]

So far we have BASE or 714 and then SUBUNIT of 714 is 4. Then we write it as it is comma “as it is” minus 1. Or 4,3.

“Wrapping it around” 9 gives us: 43,9,34 where 9 is actually square of step 2, position 2.

A caveat: the grouping has changed now: It is (43)(9)(34). We definite 9+1 as 0m where 9-2 obviously gives us 7.

Where does the picture fit in? We are getting there. Since it is a BASE 4 game, we can use the glyph of 4 with the various arches representing different forms or patterns.

Before it gets further confusing, let’s harken back to the medieval lady with arches. So there are 24 different patterns for the number 2 as the SEED VALUE. Furthermore, we decide that putting the arches to the left side will give different values. So 48 different patterns.

Difficult part is memorizing all the 24 different patterns. Then one can mix and match and generate any number. Theoretically speaking of course.

For 7-digit number sequence there are 10 million permutations.  But we only have 48 different forms for each number and there are 10 of them giving us 480.  This barely scratches the surface.

Is there a taxonomy of forms of 7-digit phone numbers? Can we at least force a pattern? The thing is each glyphs by themselves represent all of the same digits. So the Egyptian scribe, if written by itself, gives us 111-1111.

Adding the thinly bordered ogee arch of the first character in the spectrum gives us a random pattern of an algorithm.  Basically what it’s saying is take BASE-1 and perform a familiar operation. And when it’s on the left, it will perform a reverse operation.

Admittedly of course, the ornamental arch may not be quickly grasped driving at 65mph highway; but, due to the visual nature it will stick to memory more easily.

Another possible representation can be using the skeletal form of the following:

Bubbling in the nodes will give either number. Open or closed. And shading in segment will give a sequence.  Do visual tangrams exist for all permutations of 7-digit number? How about 123-4567? May be an acyclic graph for 1/2?

We have reached a cul-de-sac. Introducing a new glyph, hopefully will help resolve the missing pieces of number forms.


Can a mathematician predict a car accident? Wednesday, Jan 25 2012 

May be it’s because of my recent circumstances, but Traffic Flow Theory (informally the mathematics of traffic flow) has always interested me.  A monograph on this subject can be accessed here and the second page in Chapter 4 has some useful notations relating to this matter. Although mathematicians always enjoyed a prophetic ability of prediction- whether it be predicting an eclipse or the next perfect number- they cannot still harness their powers in plethora of avenues such as seismic events’ precognition.

Thus I ponder how much can we lasso the focus to scrawl an equation on blackboard that will enable predicting a car accident before it happens. Surprisingly, Google seemed to have a good match for the searches and it took me precisely to a link which has discussed this issue: The numbers behind NUMB3RS: solving crime with mathematics

Now admittedly, I am not a big fan of the show, simply because I feel that it has tried too hard to popularize the sober art of mathematics for lay audience. Not everything must be edutainment and while Scott Free productions has some excellent visuals,  it has very much watered down the mathematics insulting audience intelligence. But my intention was not a critique of the show and it seems there is an episode which aired titled Manhunt that dealt with Bayesian Inference.

As the afore-cited free link of Google Books shows that apparently 9/11 attack on the Pentagon was predicted on May 2001 via a software called Site Profiler utilizing Bayesian network (also known as belief network).  The term Bayesian is named after Reverend Thomas Bayes, a Presbyterian minister, who composed two works in his lifetime, the first one of which was on theology Divine Benevolence, or an Attempt to Prove That the Principal End of the Divine Providence and Government is the Happiness of His Creatures (1731).

Bayes’ Theorem which has wide applications from Artificial Intelligence to drug testing, figuring out guilty party in taxicab accident (see here) to aiding entomologist identify a “rare subspecies of beetle” (frequentist example) can be best appreciated with the following example (Wikipedia):

If someone told you he had a nice conversation in the train, the probability it was a woman he spoke with is 50%. If he told you the person he spoke to was going to visit a quilt exhibition, it is far more likely than 50% it is a woman. Call W the event he spoke to a woman, and Q the event “a visitor of the quilt exhibition”. Then: P(W) = 0.50, but with the knowledge of Q the updated value is P(W | Q) that may be calculated with Bayes’ formula as: P(W|Q)=\frac{P(Q|W)P(W)}{P(Q)}=\frac{P(Q|W)P(W)}{P(Q|W)P(W)+P(Q|M)P(M)}

in which M (man) is the complement of W. As P(M) = P(W) = 0.5 and P(Q | W) > > P(Q | M), the updated value will be quite close to 1.

Returning to the original topic, can Bayesian inference from a mathematical model aid in assisting a car accident before it happens? Research does not predict accidents at will; they only reconstructs model of past accidents.  Journal of Transportation and Statistics has a paper by Marjan Simoncic of Slovenia titled A Bayesian Network Model of Two-Car Accidents along with Gary A. Davis’s Bayesian reconstruction of traffic accidents, the abstract of which has been reproduced below from Oxford Journal:

Traffic accident reconstruction has been defined as the effort to determine, from whatever evidence is available, how an accident happened. Traffic accident reconstruction can be treated as a problem in uncertain reasoning about a particular event, and developments in modeling uncertain reasoning for artificial intelligence can be applied to this problem. Physical principles can usually be used to develop a structural model of the accident and this model, together with an expert assessment of prior uncertainty regarding the accident’s initial conditions, can be represented as a Bayesian network. Posterior probabilities for the accident’s initial conditions, given evidence collected at the accident scene, can then be computed by updating the Bayesian network. Using a possible worlds semantics, truth conditions for counterfactual claims about the accident can be defined and used to rigorously implement a ‘but for’ test of whether or not a speed limit violation could be considered a cause of an accident. The logic of this approach is illustrated for a simplified version of a vehicle/pedestrian accident, and then the approach is applied to four actual accidents.

Clearly the question posed in the heading doesn’t seem that much in the realm of sci-fi as given by first impression. For an additional resource see the paper
PREDICTION UNDER BAYESIAN APPROACH OF CAR ACCIDENTS IN URBAN INTERSECTIONS

Hyped Up Tuesday, Jan 24 2012 

No it’s not the Redline talking. What is hyper in terms of mathematical context? Suppose Alice lives in Universe A and Bob in parallel universe B. If  Alice draws a box around a stick figure of Bob and Bob the same of Alice, does it make sense to ask if set A contains set B or vice versa?

Is there a generalization of ‘hyper’ such as if hyper in hyper-Mahlo cardinal same as hyper of hypercohomology? Or hypergame paradox for that matter.

Going back to the original discussion we will shortly introduce Zwicker’s hypergame paradox. But first let me give you this question of MathOverflow: Can we have A={A}?

My answer -the one with the lowest vote- points to Ben Goertzel’s link shows that the concept of non-well founded set theory has also been given serious thought in Situation Theory of Barwise(& Etchemendy).  Wikipedia proclaims:

Within this framework, it can be shown that the so-called Quine atom, formally defined by Q={Q}, exists and is unique.

Borrowing from here we  get a feel for how hypergame works:

A finite game is a two-player game which is guaranteed to end after a finite number of moves. It is ok for a finite game to have an infinite number of possibilities; for example, the game where player one names any number and immediately wins is a finite game. It is also ok for a game to have an unbounded number of moves, as long as it always ends after finitely many. For example, the game where player one names any number N, then N moves pass with nothing happening before player one wins. This game is still a finite game, since player one will eventually win in a finite number of moves no matter which N was chosen.

To play hypergame, player one names any finite game. Player two then makes the first move in that game, and play continues as usual. Whoever wins the game also wins the hypergame.

To draw an example suppose Alice starts her first legal move

Let’s play a hypergame.

to which Bob replies

Let’s play a hypergame.

giving the following structure

Let’s play a hypergame.

Let’s play a hypergame.

begging the obvious question if hypergame is finite or not. More interestingly if one takes a pot and throw in Ehrenfeucht–Fraïssé, Axiom of Determinacy and Newcomb’s Paradox, what stew will she be able to concoct?

Hyped up yet? Well the game has just began! Here are some links to stay busy for a while: