Recently, my fellow students and I have been caught up in the craze that is sweeping through the users of Professor B------'s clockwork calculating engine; namely the charting of sets of two dimensional points that have fractal planar boundaries, being those that in some sense have a fractional dimension. Of particular interest have been the results of repeated applications of quadratic functions to complex numbers; specifically in measuring how quickly, if at all, they escape a region surrounding the starting point, by which charts may be constructed that many of the collegiate consider so delightful as to constitute art painted by mathematics itself!
Category: recreational mathematics
Finally On A Clockwork Contagion – student
Over the course of the year my fellow students and I have spent our free time building mathematical models of the spread of disease, initially assuming that upon contracting the infection a person would immediately and forever be infectious, then adding periods of incubation and recovery before finally introducing the concept of location whereby the proximate are significantly more likely to interact than the distant and examining the consequences for a population distributed between several disparate villages.
Whilst it is most certainly the case that this was more reasonable than assuming entirely random encounters it failed to take into account the fact that folk should have a much greater proclivity to meet with their friends, family and colleagues than with their neighbours and it is upon this deficiency that we have concentrated our most recent efforts.
Whilst it is most certainly the case that this was more reasonable than assuming entirely random encounters it failed to take into account the fact that folk should have a much greater proclivity to meet with their friends, family and colleagues than with their neighbours and it is upon this deficiency that we have concentrated our most recent efforts.
Further Still On A Clockwork Contagion – student
My fellow students and I have spent the past several months attempting to build a mathematical model of the spread of disease, our interest in the subject having been piqued whilst we were confined to our halls of residence during the epidemic that beset us upon the dawn of the year. Having commenced with the assumption that those who became infected would be infectious immediately and in perpetuity we refined our model by adding a non-infectious period of incubation and a finite period of illness, after which sufferers should recover with consequent immunity and absence of infectiousness.
A fundamental weakness in our model that we have lately sought to address is the presumption that individuals might initiate contact with other members of the population entirely by chance when it is far more likely that they should interact with those in their immediate vicinity. It is upon our first attempt at correcting this deficiency that I should now like to report.
A fundamental weakness in our model that we have lately sought to address is the presumption that individuals might initiate contact with other members of the population entirely by chance when it is far more likely that they should interact with those in their immediate vicinity. It is upon our first attempt at correcting this deficiency that I should now like to report.
Further On A Clockwork Contagion – student
When last we spoke, I told you of my fellow students' and my first attempt at employing Professor B------'s wondrous computational engine to investigate the statistical properties of the spread of disease; a subject that we had become most curious about whilst confined to our quarters during the epidemic earlier this year. You will no doubt recall that our model assumed that once someone became infected their infectiousness would persist indefinitely, which is quite contrary to the nature of the outbreak. We have since added incubation, recovery and immunity and it is upon these refinements that I shall now report.
On A Clockwork Contagion – student
During the recent epidemic, my fellow students and I had plenty of time upon our hands due to the closure of the taverns, theatres and gambling houses at which we would typically while away our evenings and the Dean's subsequent edict restricting us to halls. We naturally set to thinking upon the nature of the disease's transmission and, once the Dean relaxed our confinement, we returned to our college determined to employ Professor B------'s incredible mathematical machine to investigate the probabilistic nature of contagion.
Finally On A Very Cellular Process – student
Over the course of the year my fellow students and I have been utilising our free time to explore the behaviour of cellular automata, which are mechanistic processes that crudely approximate the lives and deaths of unicellular creatures such as amoebas. Specifically, they are comprised of unending lines of boxes, some of which contain cells that are destined to live, dive and reproduce according to the occupancy of their neighbours.
Most recently we have seen how we can categorise automata by the manner in which their populations evolve from a primordial state of each box having equal chances of containing or not containing a cell, be they uniform, constant, cyclical, migratory, random or strange. It is the latter of these, which contain arrangements of cells that interact with each other in complicated fashions, that has lately consumed our attention and I shall now report upon our findings.
Most recently we have seen how we can categorise automata by the manner in which their populations evolve from a primordial state of each box having equal chances of containing or not containing a cell, be they uniform, constant, cyclical, migratory, random or strange. It is the latter of these, which contain arrangements of cells that interact with each other in complicated fashions, that has lately consumed our attention and I shall now report upon our findings.
Further Still On A Very Cellular Process – student
My fellow students and I have lately been spending our spare time experimenting with cellular automata, which are simple mathematical models of single celled creatures such as amoebas, governing their survival and reproduction from one generation to the next according to the population of their neighbourhoods. In particular, we have been considering an infinite line of boxes, some of which contain living cells, together with rules that specify whether or not a box will be populated in the next generation according to its, its left hand neighbour's and its right hand neighbour's contents in the current generation.
We have found that for many such automata we can figure the contents of the boxes in any generation that evolved from a single cell directly, in a few cases from the oddness or evenness of elements in the rows of Pascal's triangle and the related trinomial triangle, and in several others from the digits in terms of sequences of binary fractions.
We have since turned our attention to the evolution of generations from multiple cells rather then one; specifically, from an initial generation in which each box has an even chance of containing a cell or not.
We have found that for many such automata we can figure the contents of the boxes in any generation that evolved from a single cell directly, in a few cases from the oddness or evenness of elements in the rows of Pascal's triangle and the related trinomial triangle, and in several others from the digits in terms of sequences of binary fractions.
We have since turned our attention to the evolution of generations from multiple cells rather then one; specifically, from an initial generation in which each box has an even chance of containing a cell or not.
Further On A Very Cellular Process – student
You will no doubt recall my telling you of my fellow students' and my latest pastime of employing Professor B------'s Experimental Clockwork Mathematical Apparatus to explore the behaviours of cellular automata, which may be thought of as simplistic mathematical simulacra of animalcules such as amoebas.
Specifically, if we put together an infinite line of imaginary boxes, some of which are empty and some of which contain living cells, then we can define a set of rules to determine whether or not a box will contain a cell in the next generation depending upon its own, its left and its right neighbours contents in the current one.
Specifically, if we put together an infinite line of imaginary boxes, some of which are empty and some of which contain living cells, then we can define a set of rules to determine whether or not a box will contain a cell in the next generation depending upon its own, its left and its right neighbours contents in the current one.
On A Very Cellular Process – student
Recently my fellow students and I have been spending our free time using Professor B------'s remarkable calculating engine to experiment with cellular automata, being mathematical contrivances that might be thought of as crude models of the lives of those most humble of creatures; amoebas. In their simplest form they are unending lines of boxes, some of which contain a living cell that at each generation will live, die or reproduce according to the contents of its neighbouring boxes. For example, we might say that each cell divides and its two offspring migrate to the left and right, dying if they encounter another cell's progeny.
Finally On An Ethereal Orrery – student
Over the course of the year, my fellow students and I have been experimenting with an ethereal orrery which models the motion of heavenly bodies using nought but Sir N-----'s laws of gravitation and motion. Whilst the consequences of those laws are not generally subject to solution by mathematical reckoning, we were able to approximate them with a scheme that admitted errors of the order of the sixth power of the steps in time by which we advanced the positions of those bodies.
We have thus far employed it to model the solar system itself, uniformly distributed bodies of matter and the accretion of bodies that are close to Earth's orbit about the Sun. Whilst we were most satisfied by its behaviour, I should now like to report upon an altogether more surprising consequence of its engine's action.
We have thus far employed it to model the solar system itself, uniformly distributed bodies of matter and the accretion of bodies that are close to Earth's orbit about the Sun. Whilst we were most satisfied by its behaviour, I should now like to report upon an altogether more surprising consequence of its engine's action.
Further Still On An Ethereal Orrery – student
Recently, my fellow students and I constructed a mathematical orrery which modelled the motion of heavenly bodies employing Sir N-----'s laws of gravitation and motion, rather than clockwork, as its engine. Those laws state that bodies are attracted toward each other with a force proportional to the product of their masses divided by the square of the distance between them, that a body will remain at rest or in constant motion unless a force acts upon it, that if a force acts upon it then it will be accelerated in the direction of that force at a rate proportional to its strength divided by its mass and that, if so, it will reciprocate with an opposing force of equal strength.
Its operation was most satisfactory, which set us to wondering whether we might use its engine to investigate the motions of entirely hypothetical arrangements of heavenly bodies and I should now like to report upon our progress in doing so.
Its operation was most satisfactory, which set us to wondering whether we might use its engine to investigate the motions of entirely hypothetical arrangements of heavenly bodies and I should now like to report upon our progress in doing so.
Further On An Ethereal Orrery – student
Last time we met we spoke of my fellow students' and my interest in constructing a model of the motion of heavenly bodies using mathematical formulae in the place of brass. In particular we have sought to do so from first principals using Sir N-----'s law of universal gravitation, which states that the force attracting two bodies is proportional to the product of their masses divided by the square of the distance between them, and his laws of motion, which state that a body will remain at rest or in constant motion unless a force acts upon it, that it will be accelerated in the direction of that force at a rate proportional to its magnitude divided the body's mass and that a force acting upon it will be met with an equal force in the opposite direction.
Whilst Sir N----- showed that a pair of bodies traversed conic sections under gravity, being those curves that arise from the intersection of planes with cones, the general case of several bodies has proved utterly resistant to mathematical reckoning. We must therefore approximate the equations of motion and I shall now report on our first attempt at doing so.
Whilst Sir N----- showed that a pair of bodies traversed conic sections under gravity, being those curves that arise from the intersection of planes with cones, the general case of several bodies has proved utterly resistant to mathematical reckoning. We must therefore approximate the equations of motion and I shall now report on our first attempt at doing so.
On An Ethereal Orrery – student
My fellow students and I have lately been wondering whether we might be able to employ Professor B------'s Experimental Clockwork Mathematical Apparatus to fashion an ethereal orrery, making a model of the heavenly bodies with equations rather than brass.
In particular we have been curious as to whether we might construct such a model using nought but Sir N-----'s law of universal gravitation, which posits that those bodies are attracted to one another with a force that is proportional to the product of their masses divided by the square of the distance between them, and laws of motion, which posit that a body will remain at rest or move with constant velocity if no force acts upon it, that if a force acts upon it then it will be accelerated at a rate proportional to that force divided by its mass in the direction of that force and that it in return exerts a force of equal strength in the opposite direction.
In particular we have been curious as to whether we might construct such a model using nought but Sir N-----'s law of universal gravitation, which posits that those bodies are attracted to one another with a force that is proportional to the product of their masses divided by the square of the distance between them, and laws of motion, which posit that a body will remain at rest or move with constant velocity if no force acts upon it, that if a force acts upon it then it will be accelerated at a rate proportional to that force divided by its mass in the direction of that force and that it in return exerts a force of equal strength in the opposite direction.
Finally On Natural Analogarithms – student
Over the course of the year my fellow students and I have spent much of our spare time investigating the properties of the set of infinite dimensional vectors associated with the roots of rational numbers by way of the former's elements being the powers to which the latter's prime factors are raised, which we have dubbed ℓ-space.
We proceeded to define functions of such numbers by applying operations of linear algebra to their ℓ-space vectors; firstly with their magnitudes and secondly with their inner products. This time, I shall report upon our explorations of the last operation that we have taken into consideration; the products of matrices and vectors.
We proceeded to define functions of such numbers by applying operations of linear algebra to their ℓ-space vectors; firstly with their magnitudes and secondly with their inner products. This time, I shall report upon our explorations of the last operation that we have taken into consideration; the products of matrices and vectors.
Further Still On Natural Analogarithms – student
For several months now my fellow students and I have been exploring ℓ-space, being the set of infinite dimensional vectors whose elements are the powers of the prime factors of the roots of rational numbers, which we chanced upon whilst attempting to define a rational valued logarithmic function for such numbers.
We have seen how we might define functions of roots of rationals employing the magnitude of their associated ℓ-space vectors and that the iterative computation of such functions may yield cyclical sequences, although we conspicuously failed to figure a tidy mathematical rule governing their lengths.
The magnitude is not the only operation of linear algebra that we might bring to bear upon such roots, however, and we have lately busied ourselves investigating another.
We have seen how we might define functions of roots of rationals employing the magnitude of their associated ℓ-space vectors and that the iterative computation of such functions may yield cyclical sequences, although we conspicuously failed to figure a tidy mathematical rule governing their lengths.
The magnitude is not the only operation of linear algebra that we might bring to bear upon such roots, however, and we have lately busied ourselves investigating another.
Further On Natural Analogarithms – student
My fellow students and I have of late been thinking upon an equivalence between the roots of rational numbers and an infinite dimensional rational vector space, which we have named ℓ-space, that we discovered whilst defining analogues of logarithms that were expressed purely in terms of rationals.
We were particularly intrigued by the possibility of defining functions of such numbers by applying linear algebra operations to their associated vectors, which we began with a brief consideration of that given by their magnitudes. We have subsequently spent some time further exploring its properties and it is upon our findings that I shall now report.
We were particularly intrigued by the possibility of defining functions of such numbers by applying linear algebra operations to their associated vectors, which we began with a brief consideration of that given by their magnitudes. We have subsequently spent some time further exploring its properties and it is upon our findings that I shall now report.
On Natural Analogarithms – student
Last year my fellow students and I spent a goodly portion of our free time considering the similarities of the relationships between sequences and series and those between derivatives and integrals. During the course of our investigations we deduced a sequence form of the exponential function e^{x}, which stands alone in satisfying the equations
D f = f
f(0) = 1
where D is the differential operator, producing the derivative of the function to which it is applied.
This set us to wondering whether or not we might endeavour to find a discrete analogue of its inverse, the natural logarithmln x , albeit in the sense of being expressed in terms of integers rather than being defined by equations involving sequences and series.
where D is the differential operator, producing the derivative of the function to which it is applied.
This set us to wondering whether or not we might endeavour to find a discrete analogue of its inverse, the natural logarithm
Finally On A Calculus Of Differences – student
My fellow students and I have spent much of our spare time this past year investigating the similarities between the calculus of functions and that of sequences, which we have defined for a sequence s_{n} with the differential operator
Δ s_{n} = s_{n} - s_{n-1}
and the integral operator
where Σ is the summation sign, adopting the convention that terms with non-positive indices equate to zero.
We have thus far discovered how to differentiate and integrate monomial sequences, found product and quotient rules for differentiation, a rule of integration by parts and figured solutions to some familiar-looking differential equations, all of which bear a striking resemblance to their counterparts for functions. To conclude our investigation, we decided to try to find an analogue of Taylor's theorem for sequences.
and the integral operator
n | |||
Δ^{-1} s_{n} = | Σ | s_{i} | |
i = 1 |
We have thus far discovered how to differentiate and integrate monomial sequences, found product and quotient rules for differentiation, a rule of integration by parts and figured solutions to some familiar-looking differential equations, all of which bear a striking resemblance to their counterparts for functions. To conclude our investigation, we decided to try to find an analogue of Taylor's theorem for sequences.
Further Still On A Calculus Of Differences – student
For some time now my fellow students and I have been whiling away our spare time considering the similarities of the relationships between sequences and series and those between the derivatives and integrals of functions. Having defined differential and integral operators for a sequence s_{n} with
Δ s_{n} = s_{n} - s_{n-1}
and
where Σ is the summation sign, we found analogues for the product rule, the quotient rule and the rule of integration by parts, as well as formulae for the derivatives and integrals of monomial sequences, being those whose terms are non-negative integer powers of their indices, and higher order, or repeated, derivatives and integrals in general.
We have since spent some time considering how we might solve equations relating sequences to their derivatives, known as differential equations when involving functions, and it is upon our findings that I shall now report.
and
n | |||
Δ^{-1} s_{n} = | Σ | s_{i} | |
i = 1 |
We have since spent some time considering how we might solve equations relating sequences to their derivatives, known as differential equations when involving functions, and it is upon our findings that I shall now report.
Further On A Calculus Of Differences – student
As I have previously reported, my fellow students and I have found our curiosity drawn to the calculus of sequences, in which we define analogues of the derivatives and integrals of functions for a sequence s_{n} with the operators
Δ s_{n} = s_{n} - s_{n-1}
and
respectively, where Σ is the summation sign, for which we interpret all non-positively indexed elements as zero.
I have already spoken of the many and several fascinating similarities that we have found between the derivatives of sequences and those of functions and shall now describe those of their integrals, upon which we have spent quite some mental effort these last few months.
and
n | |||
Δ^{-1} s_{n} = | Σ | s_{i} | |
i = 1 |
I have already spoken of the many and several fascinating similarities that we have found between the derivatives of sequences and those of functions and shall now describe those of their integrals, upon which we have spent quite some mental effort these last few months.
Further On A Calculus Of Differences – student
As I have previously reported, my fellow students and I have found our curiosity drawn to the calculus of sequences, in which we define analogues of the derivatives and integrals of functions for a sequence s_{n} with the operators
Δ s_{n} = s_{n} - s_{n-1}
and
respectively, where Σ is the summation sign, for which we interpret all non-positively indexed elements as zero.
I have already spoken of the many and several fascinating similarities that we have found between the derivatives of sequences and those of functions and shall now describe those of their integrals, upon which we have spent quite some mental effort these last few months.
and
n | |||
Δ^{-1} s_{n} = | Σ | s_{i} | |
i = 1 |
I have already spoken of the many and several fascinating similarities that we have found between the derivatives of sequences and those of functions and shall now describe those of their integrals, upon which we have spent quite some mental effort these last few months.
On A Calculus Of Differences – student
The interest of my fellow students and I has been somewhat piqued of late by a curious similarity of the relationship between sequences and series to that between the derivatives and integrals of functions. Specifically, for a function f taking a non-negative argument x, we have
and for a sequence s having terms
s_{1}, s_{2}, s_{3}, ...
we can define a series S with terms
where Σ is the summation sign, from which we can recover the terms of the sequence with
s_{n} = S_{n} - S_{n-1}
using the convention thatS_{0} equals zero.
This similarity rather set us to wondering whether we could employ the language of calculus to reason about sequences and series.
x | |||
F(x) = | ∫ | f(x) dx | |
0 | |||
f(x) = | d | F(x) | |
dx |
and for a sequence s having terms
we can define a series S with terms
n | |||
S_{n} = s_{1} + s_{2} + s_{3} + ... + s_{n} = | Σ | s_{i} | |
i = 1 |
using the convention that
This similarity rather set us to wondering whether we could employ the language of calculus to reason about sequences and series.
On A Calculus Of Differences – student
The interest of my fellow students and I has been somewhat piqued of late by a curious similarity of the relationship between sequences and series to that between the derivatives and integrals of functions. Specifically, for a function f taking a non-negative argument x, we have
and for a sequence s having terms
s_{1}, s_{2}, s_{3}, ...
we can define a series S with terms
where Σ is the summation sign, from which we can recover the terms of the sequence with
s_{n} = S_{n} - S_{n-1}
using the convention thatS_{0} equals zero.
This similarity rather set us to wondering whether we could employ the language of calculus to reason about sequences and series.
x | |||
F(x) = | ∫ | f(x) dx | |
0 | |||
f(x) = | d | F(x) | |
dx |
and for a sequence s having terms
we can define a series S with terms
n | |||
S_{n} = s_{1} + s_{2} + s_{3} + ... + s_{n} = | Σ | s_{i} | |
i = 1 |
using the convention that
This similarity rather set us to wondering whether we could employ the language of calculus to reason about sequences and series.