Dear Madam,

This is our letter to Dr. Wiliam Mc Harris in his thread. We thought it may be of interest to you.

Mathematics is the science of accumulation and reduction of similars or partly similars. The former is linear and the later non-linear. Because of the high degree of interdependence and interconnectedness, it is no surprise that everything in the Universe is mostly non-linear. The left hand sides of all equations depict free will, as we are free to chose or change the parameters. The equality sign depicts the special conditions necessary to start the interaction. The right hand side depicts determinism, as once the parameters and special conditions are determined, the results are always predictable. Hence, irrespective of whether the initial conditions could be precisely known or not, the results are always deterministic. Even the butterfly effect would be deterministic, if we could know the changing parameters at every non-linearity. Our inability to measure does not make it chaotic - "complex, even inexplicable behavior". Statistics only provides the minimal and maximal boundaries of the various classes of reactions, but never solutions to individual interactions or developmental chains. Your example of "the deer population in Northern Michigan", is related to the interdependence and interconnectedness of the eco system. Hence it is non-linear.

Infinities are like one - without similars. But whereas the dimensions of one are fully perceived, the dimensions of infinities are not perceptible. (We have shown in many threads here without contradiction that division by zero is not infinite, but leaves a number unchanged.) We do not know the beginning or end of space (interval of objects) or time (interval of events). Hence all mathematics involving infinities are void. But they co-exist with all others - every object or event exists in space and time. Length contraction is apparent to the observer due to Doppler shift and Time dilation is apparent due to changing velocity of light in mediums with different refractive index like those of our atmosphere and outer space.

Your example of the computation of evolutionary sequence of random numbers omits an important fact. Numbers are the inherent properties of everything by which we differentiate between similars. If there are no similars, then it is one; otherwise many. Many can be 2,3,...n depending upon the sequence of perceptions leading to that number. Often it happens so fast that we do not realize it. But once the perception of many is registered in our mind, it remains as a concept in our memory and we can perceive it even without any objects. When you use "a pseudorandom number generator to generate programs consisting of (almost) random sequences of numbers", you do just that through "comparison and exchange instructions". You develop these by "inserting random minor variations, corresponding to asexual mutations; second, by 'mating' parent programs to create a child program, i.e., by splicing parts of programs together, hoping that useful instructions from each parent occasionally will be inherited and become concentrated" and repeat it "thousands upon thousands of time" till the concept covers the desired number sequences. Danny Hillis missed this reasoning. Hence he erroneously thought "evolution can produce something as simple as a sorting program which is fundamentally incomprehensible". After all, computers are GIGO. Brain and Mind are not redundant.

Much has been talked about sensory perception and memory consolidation as composed of an initial set of feature filters followed by a special class of mathematical transformations which represent the sensory inputs generating interacting wave-fronts over the entire sensory cortical area - the so-called holographic processes. It can explain the almost infinite memory. Since a hologram retains the complete details at every point of its image plane, even if a small portion of it is exposed for reconstruction, we get the entire scene, though the quality is impaired. Yet, unlike an optical hologram, the neural hologram is formed by very low frequency post-synaptic potentials providing a low information processing capacity to the neural system. Further, the distributed memory mechanisms are not recorded randomly over the entire brain matter, as there seems to be preferred locations in the brain for each sensory input.

The impulses from the various sensory apparatus are carried upwards in the dorsal column or in the anterio-lateral spinothalamic tract to the thalamus, which relays it to the cerebral cortex for its perception. At any moment, our sense organs are bombarded by a multitude of stimuli. But only one of them is given a clear channel to go up to the thalamus and then to the cerebral cortex at any instant, so that like photographic frames, we perceive one frame at an instant. Unlike the sensory apparatuses that are subject specific, this happens for all types of impulses. The agency that determines this subject neutral channel, is called mind, which is powered by the heart and lungs. Thus, after the heart stops beating, mind stops its work.

However, both for consolidation and retrieval of sensory information, the holographic model requires a coherent source which literally 'illuminates' the object or the object-projected sensory information. This may be a small source available at the site of sensory repository. For retrieval of the previously consolidated information, the same source again becomes necessary. Since the brain receives enormous information that is present for the whole life, such source should always be illuminating the required area in the brain where the sensory information is stored. Even in dream state, this source must be active, as here also local memory retrieval and experience takes place. This source is the Consciousness.

Regards,

mbasudeba@gmail.com

Dear Dr. Perez,

I'm glad you read Dr.Devereux's essay and last night I read you comments in her blog. I have also read (last week) her replies to the questions, including what she meant by space, and there she says that she means space-time-energy continuum. I find her vision (as I understand it, since it is essentially the same as mine) particularly fascinating in the sense that it has 2 inherent characteristics: it is both a continuum (in the full meaning of this term) and discrete (discreteness arises from oscillations whose period is not fixed).

.

You: "Again, the uncertainty principle and the measurement problem, mean either "interfere physically" or "place a detector". A detector is a measuring device that interferes with the system under study, in this case the atoms. When we perform a measurement, i.e., we look, we put in action the uncertainty principle and at the same time the measurement destroys the state of the system, i.e., the wave function collapses."

You can't be serious -? At first I thought you were joking but since we're not skyping and I can't see the twinkle in your eye, I can't be sure.. So, just in case you are serious, this would actually fit very well in someone's view where information and underlying reality are one and the same (which I do not believe for a second applies to you -- I think you're just pulling my leg lol).

..because the uncertainty principle is NOT the reality but a theoretical construct based on the fact that we do not have sufficient information about the system, and so we are forced to consider its all possible states before 'measurement'. However, in reality, the system IS in one of those possible states -- but which one we do not know. This is the underlying reality. The collapse of the function is the theory, or 'information' based on the lack of information. In reality, when a detector is placed in the path of an atom to determine whether it passed through the given slit, this is direct _physical_ interfering with the atom and its milieu. Nothing collapses in reality. One out of all probable states was real and it is revealed after the interaction.

.

I still did not have a chance to review the SR paradoxes -- it's on my list and I watched the video. In this regard, do you remember the essay by Andrew Norton last year? On my blog he gave very interesting links to water droplets emerging in beautifully ordered formations on the surface of vibrating medium. They last quite a long time. I did not see the video of that experiment but only the photographs from the paper. It's quite stunning (I'll see if I can find it googling 'oscillating water droplets')

.

You: "Do you get what a mean by medium?"

Yes, I surely do, even though I --in my naivete perhaps-- also differentiate in the types of.. signals? it can carry. For example, EMR is a transverse wave and gravity is supposedly longitudinal (in 4D), but in addition, it can carry the resonances of vibrations, no? I 'm not sure how to describe it.. I 'see' in my head how disjoint (not adjacent) 'chunks' of space vibrate and if they do so 'in harmony' they sort of 'hear' each other over the distance and so reinforce each other -- while, say, a 'dis-harmonic' chunk, surrounded by a vibrating entirely out of phase milieu, can be annihilated -- puff! out of existence. And 'forces' arise at the boundaries between the adjacent 'dis-harmoniously' vibrating chinks of space. (I realize this must sound wacky and that's why I prefer the professionals do the talking ;)

Sincerely,

-Marina

Dear Marina,

I want to express my best congratulations for your very well written essay, which I rated very high.

Needles to say that I share your position that "information lies not only in the heart of Life but also is at core of `reality' studied by physics''. What I'd like to remark is that it is crucial that such information be quantum. Hence: It from Qubit. Space-time would never emerge from classical bits: it needs quantum bits! The classical information is the one available to the observer, what we call the "event"-the experimental outcome-everything that we consider objective. Or else, using the cryptographic paradigm: classical information is openly known, quantum Information is secret. Regarding Wheeler's "participatory universe", I'm not sure I share this. The boundary that generates new information as mentioned in your essay is the boundary between the coherent quantum evolution and the observer, where the secret is disclosed to the yes-no question.

And to get to your finale: information is never stored. It always flows. Storing information is a temporary loop made with interaction with ancillary systems.

At least, this is my own re-reading of your essay.

Compliments again, Marina.

My best wishes

Mauro

Dear Marina,

What a lovely, lyrical essay! I enjoyed it thoroughly, for it combines common sense with scientific discipline. And I think you are skirting around the idea of nonlinear dynamics, even chaos theory, with your recursive loops. I expound on this idea a little more in my reply to your comments on my essay, "It from Bit from It from Bit..."

Again, thanks for your comments -- and especially for your insight and lovely writing style.

Best wishes,

Bill McHarris

    Hi Marina

    Indeed, some people, including myself, think that the quantization arises from the continuum and some plausible explanations have been given.

    I didn't understand what you really mean in your 3rd paragraph. From your 4th paragraph I could notice that you don't have clear the physical meaning of the uncertainty principle and the measurement problem.

    The uncertainty principle tells us, in its general formulation, that when two observables, say, A and B, do not commute it is not possible to determine their values simultaneously. The relation between two non-commuting observables is AB-BA=ih where i is the imaginary number and h is the planck constant divided by 2pi. When we work out the math we obtain that DeltaA*DeltaB=h/2, this is the expression of the uncertainty principle. Two observables that do not commute are, for example, the position X and the velocity V (or momentum P=mV) of a particle. So, according to the latter expression, we cannot measure with precision V and X simultaneously, because the measurement of V will affect the actual position of the particle and viceversa. Thus, in the case of the double slit experiment, the detector is placed to measure the X of the particle but the measurement of X affects its velocity. The more accurate the determination of X the higher the ignorance of V (and viceversa).

    Now, before the measurement, in theoretical terms, the quantum state of the system IS WELL DEFINED, but given that the measurement affects the state of the system it is said that the state collapses, that is, it becomes unknown. It's unknown because we don't know how the measurement will affect the system. Thus, that the detector "interferes" means that the detector perturbs the initial quantum state of the particle by either affecting the particle's velocity or its position (depending on what we are measuring). As you say, the particle is in one of many possible states and, according to the theory, the state before the measurement is WELL KNOWN but the measuring process destroys the knowledge of the state. Does it make sense?

    With respect to your 5th paragraph. I don't remember Andrew but I do remember the article, it's very interesting. Actually, you can see a new and beautiful video of that phenomenon here: http://www.youtube.com/watch?v=nmC0ygr08tE. The droplets are called walkers for two reasons, because the last name of the discoverer of this phenomenon is walker and because the droplets appear to walk on the liquid. In the most recent reports, researchers seem to reproduce quantum phenomena such as interference. These results suggest that the microscopic world is not different from the macroscopic one. In this analogy the droplet represents the resonances or excitations (that we see as particles) of space and the water represents the medium, i.e., the substantial space (the substance for me is matter). Again, I would like to insist and make this very clear, the water is the absolute system of reference just as space is. The velocity of the particles is defined absolutely not relative to other objects in space, as relativity affirms, but with respect to the water (i.e. space). Do you agree? Do you understand why the kind of resonating space (that you, Devereux, me and others support) implies an absolute system of reference? In relativity the velocity of the particles is defined relative to other objects because space itself is not a substance and cannot be taken as a reference system. Space in relativity is neither a fluid nor a material medium. Do you now understand why this contradicts relativity?

    You: Yes, I surely do, even though I --in my naivete perhaps-- also differentiate in the types of.. signals? it can carry. For example, EMR is a transverse wave and gravity is supposedly longitudinal (in 4D), but in addition, it can carry the resonances of vibrations, no?

    Hmm, it seems that you have a mixture of the notion of "medium" and that you use the word "milieu" as synonym of "medium" or "environment". From your comments I have the impression that you also understand "medium" as something that carries energy or information. I sense that you are considering a wave also as medium. No, I think that notion is sort of colloquial (no offence). In physics, in particular, in mechanics of continuous media (check https://en.wikipedia.org/wiki/Continuum_mechanics), when we say medium we mean a continuous distribution of a substance (usually matter), lets say like water, that extends over some region. For instance, the waves in the ocean are perturbations of water (the medium), without the medium waves cannot exist. On the contrary, Einstein argued that EMR needs no medium (the so called aether) and that EMR can move in empty space. According to him electromagnetic waves are not perturbations of space. It's as if we removed the water and left the waves alone moving without water, the waves are not perturbations of nothing, not even of space. Do you see what I mean by medium?

    Regards

    Israel

    Thank you Dr. Perez for your detailed explanations!

    I will reply asap. And thank you for the fantastic video of the 'walking droplets'. I was especially impressed by the footage where every other set of frames was removed to show how smoothly the droplet moves along the surface (this is a 3D version of my 4D vision of an atom bouncing on the 3D surface of our world, just as Andrew suggested). I found the jpg from the paper that he linked last year:

    http://ej.iop.org/images/0295-5075/94/2/20004/Full/epl13428fig1.jpg

    The (a) at the top left corner shows the formations that lasts --if I remember correctly-- several hours, in the range of 6-8h, before dissipating.

    Deat Marina,

    I noticed your last lovely message yesterday, and today it disappeared! By the way, I noticed also another message from you few weeks ago that disappeared (it was written before the one that is answered in my post). I read it in bed during the night, and the day after I thought that I had just dreamt it. Now I know that it was there. Two messages from you have been lost. Do you know what happened? Can you replace them?

    I just arrived to Chicago from Italy yesterday. I will answer in my blog tomorrow.

    My best regards

    Mauro

      Caro bellissimo professore :)

      there was a move to a new server and, according to the FQXi Administrator Brendan Foster, there was "a temporary hole in spacetime" while they did that. Brendan wrote in his blog that he believed the data was "just passing behind the black hole but not fallen into it", but it seems that some messages did get lost. Maybe they will reappear on Monday? The new server seems quicker.

      Awaiting your answers in your blog :)

      Thank you,

      -Marina

      Dear Professor McHarris,

      thank you so much for your warm comments and your favorable evaluation of my essay. I cannot express how much it means to me, coming from a distinguished professor like you. I left a lengthy reply in your blog yesterday, but it is gone now. Perhaps it will reappear on Monday? (there was a move to a new server in the last few days). If not, I'll try and repost it (but it will not be as spontaneous and fresh as the original, alas).

      Thank you again,

      -Marina

      Marina,

      A straightforward approach where defining the components makes your perspective more accessible to the reader.

      I like to see the discussion of macro and micro denoting the difference in object behavior and attributes. Many essay don't do that. In "It's Good to be the King," I am critical of this tendency, especially making the suggestion that the macro and micro world are interchangeable in terms of behavior and obfuscating the differences.

      Your comment, "the anthropic principle, in the context of which Wheeler presented the idea, implies that a high level of consciousness, as exhibited by human, is required.." mentions this important attribute of the Wheeler concept, and the Anthropic Principle in general. I too discuss consciousness, especially in terms of it not being possible at the BB and only 1 billion years after, unless imposing supernatural or metaphysical elements, like consciousness embedded in space.

      I would like to see your thoughts on my essay.

      Jim

      Having read so many insightful essays, I am probably not the only one to find that my views have crystallized, and that I can now move forward with growing confidence. I cannot exactly say who in the course of the competition was most inspiring - probably it was the continuous back and forth between so many of us. In this case, we should all be grateful to each other.

      If I may, I'd like to express some of my newer conclusions - by themselves, so to speak, and independently of the logic that justifies them; the logic is, of course, outlined in my essay.

      I now see the Cosmos as founded upon positive-negative charges: It is a binary structure and process that acquires its most elemental dimensional definition with the appearance of Hydrogen - one proton, one electron.

      There is no other interaction so fundamental and all-pervasive as this binary phenomenon: Its continuance produces our elements - which are the array of all possible inorganic variants.

      Once there exists a great enough correlation between protons and electrons - that is, once there are a great many Hydrogen atoms, and a great many other types of atoms as well - the continuing Cosmic binary process arranges them all into a new platform: Life.

      This phenomenon is quite simply inherent to a Cosmos that has reached a certain volume of particles; and like the Cosmos from which it evolves, life behaves as a binary process.

      Life therefore evolves not only by the chance events of natural selection, but also by the chance interactions of its underlying binary elements.

      This means that ultimately, DNA behaves as does the atom - each is a particle defined by, and interacting within, its distinct Vortex - or 'platform'.

      However, as the cosmic system expands, simple sensory activity is transformed into a third platform, one that is correlated with the Organic and Inorganic phenomena already in existence: This is the Sensory-Cognitive platform.

      Most significantly, the development of Sensory-Cognition into a distinct platform, or Vortex, is the event that is responsible for creating (on Earth) the Human Species - in whom the mind has acquired the dexterity to focus upon itself.

      Humans affect, and are affected by, the binary field of Sensory-Cognition: We can ask specific questions and enunciate specific answers - and we can also step back and contextualize our conclusions: That is to say, we can move beyond the specific, and create what might be termed 'Unified Binary Fields' - in the same way that the forces acting upon the Cosmos, and holding the whole structure together, simultaneously act upon its individual particles, giving them their motion and structure.

      The mind mimics the Cosmos - or more exactly, it is correlated with it.

      Thus, it transpires that the role of chance decreases with evolution, because this dual activity (by which we 'particularize' binary elements, while also unifying them into fields) clearly increases our control over the foundational binary process itself.

      This in turn signifies that we are evolving, as life in general has always done, towards a new interaction with the Cosmos.

      Clearly, the Cosmos is participatory to a far greater degree than Wheeler imagined - with the evolution of the observer continuously re-defining the system.

      You might recall the logic by which these conclusions were originally reached in my essay, and the more detailed structure that I also outline there. These elements still hold; the details stated here simply put the paradigm into a sharper focus, I believe.

      With many thanks and best wishes,

      John

      jselye@gmail.com

      Vasilyeva,

      The importance you have given to It over Bit is reflected in your statement "It is more fundamental than Bit, the latter being just the reflection of the former". Thus you have given more importance to objectivity of reality (It) than to the subjectivity of information (Bit). Similar sort of conclusion is reached by me in my essay (http://fqxi.org/community/forum/topic/1827), where I have said 'Bit comes from It, but mind can know of It only through Bit". Thanks for writing such an elegantly argued article with many bright points. So this essay deserves excellent rating.

      Best of luck,

      Sreenath

        Thank you Sreenath for your kind comments on my essay. I have read and rated your essay a while ago and am planning to leave my feedback in your blog now.

        Hello Dr. Perez!

        Sorry for the delay in replying. You put doubts into my mind and now I have to review everything I know about QT, which, thankfully, is not much (joke!) and I also read carefully the essay by Angelo Bassi, Saikat Ghosh, & Tejinder Singh "Information and the foundations of quantum theory", which speaks just about this issue. I would like very much to hear your opinion on it.

        In the meantime, I am still chewing on your very informative comment --thank you again so much for it!-- and I have a question:

        You: " before the measurement, in theoretical terms, the quantum state of the system IS WELL DEFINED"

        I thought that it was 'well-defined' in terms of probabilities -?

        And you're right I use 'milieu' 'environment' 'medium' freely, without clear definitions of meaning, even in my own head. I will get to it later.

        How about that essay by 3 professors?

        Thank you again, very much for your feedback,

        -Marina

        Marina,

        It is a thoughtful and well reasoned entry. I would like to push for what has become my particular point of focus in the course of this contest, that the concepts of energy and information form a dichotomy.

        By energy, I simply describe it as dynamic and conserved, while information is the very essence of definition. Being definition, it must be static, for if it was dynamic, it would be constantly changing and so not definitive. These two concepts exist as a dichotomy because there cannot be one without the other. Any existence energy has, must express some form, action, or interaction, thus expressing information. While information cannot exist without the interactive momentum of energy, or there would be no connections, effects or activities. Without this action, the resulting stasis would amount to a void, a thermal state of absolute zero.

        So while I agree with your defense of information as a necessary reflection of an underlaying reality, I think the particular focus of this contest on the concept of information alone, requires us to both isolate that concept as best as possible, then equally clarify what is overlooked. To me that would be this medium of energy, to the message of information.

        Regards,

        John M

          Thank you John for reading my essay and for your kind comments!

          I read your essay a while ago and only wished it was longer :)

          I agree with you that energy and information are closely linked. Without information, energy is just dynamics. Information is what gives it direction, 'purpose' and form . And yes, by exchanging information, living things exchange energy, among themselves and with their environment. The same applies, in simpler form, to everything in existence. Energy and information are 2 integral aspects of being. That's what I think, anyway :)

          Best regards,

          -Marina

          • [deleted]

          Hi Marina

          Thanks for the picture, I checked in my files and I have the article. If you look into the internet you'll watch a couple of two videos more, they are impressive.

          As you know, it's impossible to read all essays, I try to read as much as I can. I'm glad you recommended me to read that essay, it's interesting. In the first paragraphs they discuss what I just explained to you in my previous posts.

          With regards to your first question, I think that the essay of Bassi et al., answers it very well. I would like to add a few information just for your records. Quantum mechanics have several formulations. The first one was developed in 1925 by Werner Heisenberg and is known as the matrix formulation of QM. The next year Schrodinger came up with its famous wave equation and two years later he showed that both formulations are equivalent. There is third formulation with is a mixture of both I think is due to Dirac. Another version was developed by Louis de Broglie around 1927 and one more by David Bohm in 1952. These two seem to be complementary and are known as De Broglie-Bohm theory or simply Bohmian QM. By 1986 a new version was developed, named consistent QM. There are more modern versions that are aimed to derive or generalize the original versions, such as those developed by Adler, Besso, et al (CLS, Quantum dynamics, etc.). What we are taught at schools is the three first versions, they others are not widely known at the graduate or undergraduate level. However, those who study QM and its foundations know the other versions, Bohmian mechanics is relatively popular. I don't know the technical details of this theory but I understand its principles. As the Bassi's essays explains, the standard QM considers that the state of the system is well known before the measurement is made. Actually, the calculation of the energy for a given system is totally deterministic. One can calculate the possible outcomes of quantum system. The only problem is that we don't know which one we would measure. This is why we introduce theory of probability and statistics in QM. We never know what value the measuring instrument will display, but we can tell what would be the probability and by a series of measurements we corroborate those predictions.

          What they argue is that the randomness of the measurements does not necessarily come from the measurement itself. It could be that the initial conditions are not well defined (and therefore are also random) or the system does not evolve deterministically and linearly, as assumed in standard QM. Bohmian mechanics (BM) assumes that the initial conditions are not defined. The stochastic formulation assumes that the evolution of the system is nonlinear. In particular some experiments conducted at the mesoscopic scale (that is, experiments between the micro and the macroscale) can tell whether or not QM is nonlinear.

          What would be the implications of, for instance, BM? Well, at least this formulation seems to give a more clear picture of quantum phenomena that the standard QM. In BM, there are two fundamental equations, the pilot-wave equation and the known schrodinger equation. The pilot-wave equation is the equation that governs the nonlinearity of the system. The pilot wave is the wave that guides the particles in their evolution, it would be the analog of the pilot-wave that guides the walkers in the videos that I showed you.

          It seems that Bassi et al. are trying to develop a theory from which they can derive non-relativistic QM and at the same time a relativistic formulation but I need to read their papers in order to understand how they are going to do it.

          Cheers

          Israel

          Marina,

          The problem with agreement is it concludes the points of discussion. If you want a few more of my thoughts on physics, here are the last two contest entries:

          We Look at Time Backwards

          Comparing Apples to Inches

          One of the main reasons I made this so short is because it is quite difficult to really get into so many different concepts, that really giving any of these papers their full due is next to impossible. Generally many of the participants take a very broad and deep view that has developed from years of thought, distilled it down into the seven page (or is it nine?) limit and then get frustrated when others don't see what is obvious to them, when usually the others are at best able only to connect to what relates to their points of view and at worst, are only shamelessly plugging their entry. So I tried to focus on one particular point, that information is message to the medium of energy. If I had extended it, it would be to develop how I see the right brain as a form of thermostat and the left as a form of clock, but thought that might distract from the main point. I think I do go into that in one of those other papers though.

          Regards,

          John

          Hi Marina,

          Thank you for a delightful and insightful essay. You wrote:

          > This brings us to the provocative idea espoused by J. A. Wheeler in his celebrated thesis on quantum theory where he compares the workings of the universe with a computer... At first glance the idea strikes me as both limiting and impractical.

          But is it possible?

          > I find it comforting that this view is also in line with the ancient Buddhist definition of reality that states, The world emerges in the play of mind in emptiness.

          The Mind might be a lower layer in a computational cosmos, in which the physical world emerges at the top.

          > The recursive loop of the 'participatory scheme' suggests that information is continuously generated by the events, large and small, near and far; and that each event, or process, sees its own thread of causality; and together these threads weave the beads of events into the intricate tapestry of reality

          These threads of causality (could you call them observers?) each have their own "explicate" view of an "implicate" order in my essay Software Cosmos. I describe in detail how our physical world might be possible to simulate. I then conduct a test to determine whether we are in such a simulation.

          I think it may be close to the vision you describe. I hope you have a chance to take a look and let me know what you think.

          Hugh

            Dear Dr Vasilyeva,

            I like your idea that information is repeatedly re-absorbed by reality. It's a very logical argument and immediately made me think of fractals. So glad to see these mentioned.

            Original and one of my favourite essays so far!

            My essay based partly around the Fibonacci sequence will hopefully be of interest to you.

            Excellent work - well done! Top rating from me on 5th July! Please take a look at my essay if you have time.

            Antony