The parabola has the well-known property of reflecting axis-parallel rays to a point, as can be seen in this video clip. If we rotate the parabola around its axis, we create a parabolic disc. It has the well-known property of reflecting parallel rays (= planar wave fronts) to a point - if they are incident along the axis direction of the disc - which can be seen in this video clip.
Now, it is a beautiful (and little known) geometrical fact of great practical consequence that we can avoid the astronomical costs associated with creating a large parabolic disc, and harness the work-power of the sun in a more efficient way by bending two flat mirror sheets into the shape of two parabolic cylinders. If these parabolic cylinders are properly configured, the incoming parallel light rays (= planar wave fronts) will create an exact point focus after two successive reflections. This is due to
If the focal line of the first cylinder is identical to the generating line of the parabola that is the intersection of the second cylinder with a plane perpendicular to its axis, then the incoming rays will be reflected to a perfect point.
The DCPF principle was discovered on 16 November 1976 by Ambjörn Naeve while working with Lloyd Cross, a well-known physicist and holographer who co-founded the School of Holography and the Multiplex Company in San Francisco, California. The original insight was recorded on this piece of paper, and a geometric proof of the DCPF principle can be found here.
Between 9-12 December 1976, Lloyd Cross and Ambjörn Naeve built the first point-focusing mirrors based on the DCPF principle. A short description of this intense event, including some pictures, can be found here.
A computer-based animation of the double-cylindrical point-focusing mirrors in action, created by Ambjörn Naeve using the Graphing Calculator, can be seen in this video clip, and a live “burning demo” with a DCPF mirror-pair, built by Tomas Elofsson in 1989, can be seen in this video clip.
On 1 July 1989, in Gusum, Sweden, the DCPF mirror-pair shown in the latter video clip (1.5 m^2 primary mirror) was used in an experiment by Ambjörn Naeve and Tomas Elofsson that succeeded in melting limestone (2560 ˚C) in free air. In the pictures below, Tomas Elofsson is shown tuning the mirrors and melting the limestone sample:
The DC pointfocus videos by MathRehab on YouTube
1) It is easier to build in large sizes, since cylindrical surfaces are curved in only one direction and therefore can be BENT into shape from a flat sheet, hence avoiding the “astronomical costs” that occur when a doubly-curved mirror has to be CAST into a fixed shape.
2) The focal point can be positioned outside of the solar influx area between the mirrors, where it is freely available to perform work, as in the DCPF wheelbarrel design.
3) The focal point can be moved by rotating the second cylinder around the focal line of the first cylinder. This fact can be used to create various forms of solid-state heat engines, for example by successively heating and cooling a series of bi-metal coils.
4) The mirror configuration can be approximated with planar strips, the number of which grows LINEARLY with the overall size of the construction – as opposed to the parabolic disc, where the number of planar approximators grows QUADRATICALLY with construction size.
As has been demonstrated above, the DCPF provides a cheap and efficient way to create “a huge burning glass”, which can achieve very high temperatures. Therefore, it enables local (= rural) development of a multitude of technologies that presently require advanced and expensive high-tech labs, such as:
• Solar-powered steel plants that recycle scrap metal into a valuable resource for the local community. A discussion and some designs of what a solar steel plant could look like can be found here.
• Cheap and efficient super-heated external combustion engines (steam, sterling, …), whose work-power (like that of all heat engines) is directly proportional to the difference (in ˚K) between input and output temperatures.
• Highly efficient solar electricity generators, which concentrate the energy on high-performance photo-voltaic cells that must be water-cooled to avoid melting.
The story (some parts of which are in Swedish) of 25 years of trying to get people interested in the double-cylindrical point focus - from its discovery in 1976 and up until 2001 - can be found here. One of the chapters in this story took place in July 1995 in Gunnarskog in Värmland/Sweden, where Ambjörn Naeve and Tomas Elofsson recorded a video with their daughters acting as solar energy consultants, helping a farmer to melt horse shoes and load batteries by concentrating light on a solar panel.
These activities created a big stir in the community, with the local newspaper Värmlandsbygden publishing a cover story on the DCPF with a frontpage headline stating: “They have the solar power that melts mountains” (“De har solkraften som smälter berg”). Some local politicians got involved, interested in creating jobs for the local community, and it looked like things would start moving when an application for funding was sent to the Swedish Innovation and Industry Development Agency (NUTEK), which at that time was responsible for strengthening regional growth by furthering industrial applications of innovations. However, the application was dismissed on grounds that included the following reasons (translated from Swedish):
“A problem with the suggested design is that the temperature becomes very high. This means that the heat losses are increased, which makes the [DCPF] technique less interesting in energy applications. Moreover, concentration of light by mirrors is part of the research that is already carried out within the solar energy domain - although with different configurations than the DCPF, which lead to lower temperature levels and lower heat losses.”
By that time it had become painfully obvious - from this response as well as from other responses similar to it - that we were facing a serious pedagogical problem, and that there was a lot of learning of basic physics that needed to happen in several places in society for things to move forward.
The DCPF can contribute to the design of a more decentralized and sustainable energy technology that will be necessary as our technology must evolve beyond its present oil-burning stage.
The DCPF enables a more simple and less expensive approach to high-temperature solar energy applications that makes it possible to apply and maintain them in “low-tech” rural areas that are facing increasing difficulties of raising capital for development. For example, using the DCPF, low cost, medium temperature “scrap-mirror-strips” (that are “left-overs” from ordinary mirror production) can be used to create effective solar cookers, water-purifiers, and sea-water-desalination systems.
The DCPF is not a traditional “product” but a piece of globally important information for the sustainable survival of humanity in the rapidly emerging “post-oil-burning” age. If a big corporation wants to market solar energy concentrators, it can afford the large investments (“astronomical costs”) involved in creating a big parabolic disc reflector. However, this does not apply to smaller operators in poorer communities (such as rural areas in developing countries), who cannot afford these kinds of investments. About half of the people on this planet live under such conditions, and they are the principal beneficiaries of the DCPF. But we all benefit from a global increase of energy-independence, which can enable the Big Switch that humanity is facing for sustainable survival: Decreasing our dependence on globalized production processes, supported by localized information (secrets, patents, …) and increasing our dependence on more localized production processes supported by globalized (= globally relevant) information.
1) Create an online Interactive Learning Environment around the DCPF principle, where “solar engineer wannabes” from across the globe can learn how to apply this technique to solve local production problems. The beginnings of such an online ILE can be found here.
2) Start an “ethical branding” movement among technical universities, that can work in partnerships with rural communities and developing countries in order to field-test various materials and designs. Such testing will be crucially important in order to optimize the DCPF constructions for a variety of different local conditions.
3) Create a pool of globally relevant and open ideas that can be contributed to under clear licensing conditions. Such licensing can build on Creative Commons. The default license should be: “Free to use, with proper accreditation to the source.“
Mathematics – in the sense of the Greeks, began with Thales around 600 BC – as part of an attempt to try to understand the world in a sensible (= non-magical) way. In fact, it was a part of a “scientific awakening” of the human mind against the tyranny of the priests. People started asking questions such as “what is the basic stuff that the universe is made of” – and asking such questions, not to the gods and religious authorities, but to nature itself. This was the birth of the scientific process.
/////// a few words about the difference between mathematics and science.
The word ‘mathematics’ is said to go back to Pythagoras (around 500 BC), who called his most advanced disciples ‘mathematikoi’. In the present context, this word will be interpreted in the following way:
Mathematics is the study of the totality of structures that the human mind is able to perceive.
Within the language of itself, mathematics can be described as the study of homomorphisms (= structure-preserving transformations) into the mind from its environment - including the mind itself.
Life is structure, and since mathematics is the language of structure, it is the language of life, the ultimate ruler of the subspace within which life can be talked and reasoned about.
Hence, mathematics is concerned with the effective and efficient representation of structure. The stronger (= more effective and efficient) the representation, the shorter the computation that has to be performed in order to achieve a certain result. This shortness is referred to by mathematicians as “mathematical elegance.”
A mathematician can be defined as a person who thinks for a day in order to avoid calculating for an hour. Yet, during the early parts of math education, mathematics is presented almost exclusively in terms of calculations, with increasingly fatal results with regard to the initial interest that almost every child shows towards the subject. The overdose of calculations that the kids are exposed to at school eventually fosters the attitude of trying to avoid calculations – but unfortunately at the expense of trying to avoid mathematics altogether. This is
Mathematics can never be taught. It can only be given the opportunity to grow. Mathematics is a feeling, a sensitivity and an awareness of structure, which is planted in each and every one of us, like a seed of the cosmic consciousness. To practice the art of mathematics is to be involved in a purely mental process, which has surprisingly strong connections with the surrounding physical reality.
When you are developing your mathematical understanding, both halves of your brain work together in the process of constructing combinations of mental fantasies that are tested for logical consistency. The right half of the brain is fantasizing, and the left part is analyzing and testing the logic of the suggested ideas. Only the ideas that survive the logical examination are elevated to the status of mathematical truths. Mathematics can therefore be described as logically tested fantasies. It offers powerful means for its devotees to overcome some of their sensory limitations and contemplate the inner profundity of the structure of the universe.
The First Class Mathematics Project aims to convey an image of mathematics as such a double-brained mental activity. In order to develop the ability for structural thinking, it is of the utmost importance that all students are encouraged and supported in their attempts to develop their own mathematical fantasies during their entire learning process. This is only possible if they are confronted with interesting examples of ‘good mathematics’ during every stage of their mathematical education. An illustrative example of the effects of such a confrontation is given by the so called Rubrik´s cube (known simply as “the cube“), which was such a fascinating mental torment to the kids in the 1980s. They got into doing their own advanced forms of algorithmic mathematics - but only during their breaks and time of leisure from a school-system that was totally incapable to realize what was going on, let alone to make use of it in order to promote the mathematical interest of the students.
By the aid of modern computers, many exiting mathematical structures can be animated and brought to interactive life in ways that open new and exiting pedagogical possibilities. Today there is a large number of computer-based mathematical tools that empower a student to explore mathematical concepts in interactive and dialectical ways, i.e. in an ongoing dialogue with the computer program (and of course with the teacher). It is the aim of the First Class Mathematics project to contribute to the developments of such tools, as well as to promote a pedagogical strategy that makes it possible to integrate them in the educational process in a way that stimulates interest to learn more about the underlying mathematical structures.
Mathematics has fought a long and hard historical battle to rid itself of meaning and express its content as pure form. As expressed by the great mathematician Hermann Weyl in the beginning of the last century:
Meaning is brought back into mathematics by applying it to something, which means interpreting the symbols. Hence applied mathematics can be seen as a transformation process from form to meaning.
However, because of the way the educational systems are structured, we normally fail to explain to students the meaning of being meaningless, which is that we can capture many different meanings in the same form. If, whenever we write a mathematical expression on the board, we were to give at least two different interpretations of the symbols, then this lack of understanding would not appear. As things are, we foster it from the start.
Although well-known to mathematicians and scientists, this fundamental systemic loop between meaning and form is rarely spoken of in math education? Why is that so? There are many underlying causes, but a major reason is that we have a tradition – going back to the old Greeks - of dividing mathematics into “pure” and “non-pure” (= “applied”). These two realms are normally handled by different departments with very different perspectives on mathematics.
Moreover, there is a huge bureaucratic structure in education that could be called “the difficultification industry,” which makes a living from organizing the mathematical curriculum in a way that emphasizes the difficulties – in effect telling learners that most mathematical concepts are too difficult to address at an early age, because before you start with concept/subject C you have to master concept/subject B and before you begin with B you have to master concept/subject A. This linear view on curricularized education in effect kills the highly non-linear sparks of curiosity and interest that are the most powerful drivers of generative learning.
A good example of this is given by the so called Rubrik’s cube, a “mathematical bomb” which exploded on the youth community back in the 1980s. Little kids, driven by a glowing interest, were busy inventing their own algorithms, creating and exploring mathematical structures. And how did the school system respond to this golden mathematical opportunity? In Sweden, the “difficultification industry” outlawed it, and came up with a rule that the kids were not allowed to play with the cube in school. Imagine what could have happened if instead they had exploited this wave of mathematical interest and brought in people with mathematical knowledge that could have started to explain some of the mathematics behind the cube. But that type of disruptive event has no place within the linear curriculum system.
Children have no problems learning language abstractions, neither do they fail to see the advantage of using such abstractions. No language teacher would dream of telling a pupil that you cannot yet use the word “vehicle” because it’s too abstract at this stage, you must talk about “meaningful” things such as “car,” “boat,” and “bicycle.” Yet, this happens all the time within the difficultified math education process.
Neither do children have problems with learning abstract and meaningless games – in fact they love to do so. Chess is a good example. No one outside of the difficultification industry would ever dream of telling a kid that “chess is too abstract for you, you should not play it until you are 15 years old.” Also, no chess-playing kid ever runs into problems like: “I don’t understand chess because I don’t understand why the bishop is moving diagonally.” Yet, within the difficultified math education process, questions like this are being asked all the time, because, inspired by the anti-abstract and “meaningful” approach of “week-day mathematics,” meaning is being desperately sought for in the wrong places.
Sadly, people with real knowledge of mathematics are increasingly rare within primary and secondary schools. In fact, many people who did not understand (or were not even exposed to) the mathematical material presented during the later stages of math education are recruited to teach in the early stages, i.e., they teach mathematics for young kids. This has catastrophic consequences for maintaining and stimulating the interest in mathematics among young people, on which the future of our high-tech society ultimately rests.
Alan Kay - one of the early pioneers of the graphics work station computer, and a long-term pedagogical innovator in the field of children and mathematics - has said that “The kids can literally smell the fear of the math teacher in front of them.” This fear stems from the traditional teacher role as a “knowledge filter”: I’m going to teach you what I know. In practice, this knowledge filtering strategy is almost invariably translated into carrying out a large number of very similar calculations from a book. This didactical strategy could be termed the Goebbels’ principle of math education: If you carry out a calculation a large enough number of times, then you will learn something from it.
In fact, it is the exact opposite. The will to understand what is going on is gradually turned into mindless mechanical repetition. An excellent example of this was given by my daughter Ylva, who came home from school one day (when she was in fifth grade) and said:
It is interesting to observe that many “math-damaged” people can remember exactly the point in time and the situation that made them give up on understanding math. A common “weapon of math destruction” used by scared knowledge-filtering teachers is to ridicule pupils that ask questions they do not know how to answer. The classical example of such a question is the following: “Why is minus times minus equal to plus?” The typical (fearful) response of such a teacher makes the pupil feel that s/he has asked a stupid question, and that s/he is the only one in class that “hasn’t gotten it.” In fact, s/he may be the only one in class that realizes that this question actually needs a sensible answer.
How can we change this destructive pattern? One way would be to bring in early math teachers that really know the subject. However, this is not a realistic alternative for several reasons, the main one being the low status and low salary that is associated with early teaching. People that know mathematics and know how to communicate that knowledge are eagerly sought after and much more highly paid elsewhere.
We must transform the destructive knowledge-filtering role of the early math teacher into more of a knowledge coach: “I will help you to find out more about the tings that you are interested in.” This implies the introduction of some form of support system (mathematical helpdesk), where questions can be routed to knowledgeable people that can provide live answers and stimulating conversation around the subject.
During the early stages of math education, we focus only on meaning. In Sweden there is a concept called “weekday-mathematics” (“vardagsmatematik”) (German: “Alltagsmatematik”), which tries to discover the structures.
We desperately need to take a complementary approach. We need “week-end mathematics”, the kind of math that fascinates people and engages them in conversation – the kind of math that you can talk about in the pub on Saturday night – the kind of math that can get you laid. Is there such a kind of math? Yes there is. It’s the math of many dimensions, of curved spaces, the math that Einstein used and that Cameron Diaz dreamed about understanding, something that resulted in the book E = mc2. Of course there is such a kind of math. That’s why people become mathematicians. It’s just that we are so terribly bad at bringing it to kids.
And why is that so? Basically because we suffer from the tradition that we have to be able to prove everything that we show. And mathematical proof has become so hard to teach that it has been mostly taken out of the curriculum. So we need to bring in “mathematical show” as early as possible, something that can be done by making use of modern computer simulations and interactions.
The Problem/Solution/Elimination pattern is concerned with the difference between solving and eliminating problems. Solving problems has the effect of making their symptoms dissolve, which means to ‘disappear in solution’ - just like salt-crystals in water. However, from this solution the symptoms of the problem can be easily crystallized in various ways. This tends to crystallize (= institutionalize) any organization that has been formed in order to solve problems.
Eliminating problems means dealing with causes instead of symptoms - which tends to transcend the problems as well as the organization . In fact, eliminating the problems dissolves the organization - instead of crystallizing it. Instead of becoming institutionalized, the organization ‘dissolves into solution’ - from which it can be conveniently re-crystallized if the problem should ever show up again.
The judicial system is seen as a way to solve the problems of crime by confining the criminals in jail. This activity is supported by the legal system, and the re-crystallization of convicted criminals is assured by various forms of economic dysfunctionality, most of all the natural reluctance to employ an ex-convict. Many criminals live their lives within this loop, while others organize and set up some kind of legal front in order to break the cycle. In this way the organized criminal is seen as a form of emergent evolutionary outgrowth - supported by the problem/solution pattern.
The pattern describes how conceptual difficulties (= concept-formation-problems) in mathematics education are ‘solved’ by promoting algorithmic ability, i.e. by teaching various forms of ‘arithmetic schemes’ in a more or less fundamentalist fashion. This dissolution process is modeled as driven by a pedagogical system that operates according to the principle of “behave or degrade”, i.e. the traditional math-test-metric. This leads to severe forms of understanding dysfunctionality, which in turn drives the crystallization of the conceptual difficulties. By nurturing the difficulties, the loop creates anticipated conceptual difficulties, which are summarized in the biggest mental block of all - “mathematics is difficult.” This conclusion is here modeled as an effect of the (be)cause: “I never understood it when I was at school.” This is a natural form of defense reaction that ties in with the Discriminator Questions pattern, which describes the filtering of the educational system and expresses the view that “when you do not understand during the later stages of your education, you often end up teaching in the earlier stages“.
Another variation on the problem-solution pattern appears in the later (often called higher) parts of mathematical education and research. This pattern shows the conceptual difficulties being dissolved into academic status, driven by an assessment system that operates according to the well-known principle of publish or perish.
This leads to another kind of understanding dysfunctionality - where research articles are written, not in order to be genuinely understood, but rather in order to “pee in an academic habitat,” which means to “fend of intruders” and stake an intellectual claim which is as large as possible. Moreover, this leads to the nurturing of difficulties for yet another reason - namely in order to maintain the status of the professional mathematician: Mathematics is difficult - because - I understand it, and I am smarter than you.
It does not:
• stimulate interest
• promote understanding
• support personalization
• support the transitions between the different school levels
• integrate abstraction and application
• integrate mathematics with human culture
Promoting life-long learning based on interest by:
using ICT to increase the ”cognitive contact” by:
• visualizing the concepts.
• interacting with the formulas.
• personalizing the presentation.
• routing the questions to live resources.
improving the narrative by:
• showing before proving.
• proving only when the need is evident.
• focusing on the evolutional history.
• Ideology: Within the mathemagic project we want to emphasize the speculative and creative aspects of mathematics.
• Aim: To stimulate interest in mathematics among young and old by emphasizing “week-end mathematics”.
• Basic idea: Problematize and dramatize the major mathematical concepts by anchoring them in the history of ideas.
• Methdod: Improving the narrative – e.g., by showing without necessarily proving.
• Form: The news of yesterday: Proust: “In Search of Lost Mathematics.” Knowledge components (featuring Pythagoras, Archimedes, Newton, …) are “tied together” by a ”news anchor in space-time“ who follows different trails along the evolution of mathematical ideas.
The story of the people who thought the world was understandable. From Thales and Pythagoras to Demokritos and Aristarkos.
The story of the people that wanted to escape the realm of the senses. From Plato via Augustinus and Aquino to the “scolastic age.”
The mathematics of the eye: The development of true perspective. From Pappos via de la Franchesca and da Vinci to Desargues, Pascal, Poncelét, Plücker, Grassmann and Klein.
Einstein for Flatlanders: Two-dimensional relativity theory. The story about the flatlanders that lived on a sphere and the flatlanders that lived on a torus (“dough-nut”).
The story of the people that disregarded almost everything. The evolution of abstract thinking: From induction and abduction to abstraction and deduction. “The power of thinking is knowing what not to think about.”
About the difficulties in overcoming psychological complexes. The story about the development of the concept of number: From “positive” to “negative”, from “rational” to “irrational”, from “real” to “imaginary” and “complex.”
What is there between the atoms? Does the world consist of particles or waves - or maybe something else? The historical debate from Thales versus Pythagoras via Newton versus Huygens to Einstein versus Bohr and Heisenberg and the break-up of the particle concept (super-string theory).
The mysterious law about the degradation of work: The principles of energy and entropy. The development of the energy concept from Leibniz via Rumford and Carnot to Maier, Joule and Bolzmann.
The story of the long-lived demon that was unable to forget. Maxwell’s demon and the deep connections between information theory and thermodynamics.