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Origami is the Japanese art of paper folding, (折り紙; oru - bending, kami-paper). Definition of origami says that you should fold paper without using scissors and glue, and the result of the folding is variety of forms that imitate the nature and things that are uses in everyday life. When origami is mentioned, people are usually associated with paper cranes and flowers, but it is much more due to its axiomatization. In the 20th century origami has been connected scientifically with mathematics. Humiaki Huzita designed a set of six basic ways of defining a single fold by aligning various combination of excising points, lines, and the fold line itself. These six operations are known as Huzita’s Axioms. In 2001, Koshiro Hatori added one more axiom. Huzita-Hatori axioms define what can be constructed with single fold in terms of lines and points. Some of the basic origami notions are: common shape used in origami is square, a line in origami is a crease made by a paper fold of the boundary of the paper and a point is the intersection of two lines (Alperin & Lang, 2009).
An ancient Japanese origami skill can be used in math lessons related. Folding paper into three-dimensional geometric object is a kind of an exercise. Mathematics learning through the model from the close environment of students and new methods of teaching mathematics such as discovery learning, improve teaching process and increase student interest and motivation. By folding paper students experience possible solutions, the multiform of the problem and path to accomplishment. The research has shown that origami is a beneficial method for teaching mathematics (Boakes, 2009, Robichaux and Rodrigue, 2003, Cipoletti and Wilson, 2004). It contributes developing mathematical vocabulary, special abilities and visualization. “Mathematics come alive in the mind of young students”, is said by Huse, Bluemel and Taylor (1994). Mathematical concepts can be conveyed to students by hands-on activities where the atmosphere in the classroom is more relaxed and motivating to students. Students become more curious and interested in investigating mathematical topics more profoundly (Budinski, 2009). Features of origami such as creating a model and following the procedure, spatial manipulation, generalizing procedures of different models or cooperation, application and students oriented activities are beneficial to process of learning and teaching mathematic (Meyer &Meyer, 1999). Presentation gives examples of how to use origami in teaching mathematics, on different levels, from elementary to the advanced mathematical concepts.
The fundamental problem of origami design is how to fold a square to produce a representation of desired object. The mathematics underlying origami address: existence, complexity and algorithms (Lang, 2008). There are many mathematical contents interwoven in origami and that connection influenced the mathematical education. There are more and more examples of beneficial application of origami in the classroom and some of them are supported by technology. It is an interesting combination of ancients and contemporary techniques (Fenyvesi et al, 2014). Both reach the solution successfully and comprehensibly even to average teenage student. It gives students opportunity to solve one problem in different perspectives. Combining different aspect of solving problem in the math lesson, we can improve problem solving and reasoning skills of the students. Mixed approaches to one problem can overcome weakness of a single and give students clearer picture about the problem and solution.
Origami requires following procedures of folding paper, while GeoGebra allows creating set of procedures that will lead to the solution. One hand, origami is based on one solution, while GeoGebra, on the other hand provide creation lots of examples. Also, origami requires accuracy and neatness. Combination of origami with dynamic-geometry-algebra systems, such as GeoGebra, can provide new knowledge to students. In the example constructions that combine origami and GeoGebra, we have adjusted some already known examples (Hull, 2006). Those examples are known as basic hands-on mathematics-origami, and suitable for learning various mathematical concepts. Through our examples, students solved, for example “ancient unsolvable problems” and investigated mathematical concepts that have inspired lots of mathematicians through centuries.
References and Notes
Alperin, R. and Lang, R. J. (2009). One-, two-, and multi-fold origami axioms. Origami 4, 371-393, A K Peters, Natick, MA.
Boakes, N. (2009). Origami instruction in the middle school mathematics classroom: Its impact on spatial visualization and geometry knowledge of students. Research in Middle Level Education Online, 32(7), 1-12.
Cipoletti, B., & Wilson, N. (2004). Turning origami into the language of mathematics. Mathematics Teaching in the Middle School, 10(1), 26-31.
Fenyvesi, K., Budinski, N. and Lavicza, Z. (2014). Two Solution to An Unsolvable Problem: Connecting Origami and GeoGebra in a Serbian High School, Proceedings of Bridges 2014: Mathematics, Music, Art, Architecture, Culture, 95–102.
Hull, T. (2006). Project Origami: Activities for Exploring Mathematics. A K Peters, Ltd.
Huse, V., Bluemel, N. L., & Taylor, R. H. (1994). Making connections: From paper to pop-up books. Teaching Children Mathematics, 1(1), 14–17.
Lang, R. J. (2008). From Flapping Birds to Space Telescopes: The Modern Science of Origami. Usenix Conference, Boston, MA.
Meyer, D. and Meyer, J. (1999). Teaching Mathematical Thinking through Origami. BRIDGES, Mathematical Connections in Art, Music, and Science, Winfield, Kansas.
Robichaux, R. R., & Rodrigue, P. R. (2003). Using origami to promote geometric communication. Mathematics Teaching in the Middle School, 9(4), 222-229.
Будински, Н. (2009). Оригами- пример за примену у настави математике, Педагошка Стварност, LV (9-10), 932-944.
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Future progress of new information processing devices capable of dealing with problems such as big data, Internet of things, semantic web, cognitive robotics, neuroinformatics and similar, depends on the adequate and efficient models of computation. We argue that defining computation as information transformation, and given that there is no information without representation, the dynamics of information on the fundamental level is physical/ intrinsic/ natural computation (Dodig-Crnkovic, 2011) (Dodig-Crnkovic, 2014). Intrinsic natural computation occurs on variety of levels of physical processes, such as the levels of computation of living organisms as well as designed computational devices. The present article is building on our typology of models of computation as information processing (Burgin & Dodig-Crnkovic, 2013). It is indicating future paths for the advancement of the field, expected both as a result of the development of new computational models and learning from nature how to better compute using information transformation mechanisms of intrinsic computation.
Complexity of the Concept of Computation and Information Transformation
In a variety of fields, researchers have been searching for a common definition of computation, from (Turing, 1936)(Kolmogorov, 1953)(Copeland, 1996)(Burgin, 2005) to (Denning, 2010)(Denning, 2014)(Burgin & Dodig-Crnkovic, 2011) and (Hector Zenil, 2012)(Dodig-Crnkovic & Giovagnoli, 2013). Some of these studies of computation are done in an informal setting based on hands-on and research practice, as well as on philosophical and methodological considerations. Yet other research approaches strive to build exact mathematical models to comprehensively describe computation (Denning, 2014). When the Turing machine (or Logical Computing Machine as Turing originally named his logical device) was constructed and accepted as an universal computational model, it was considered as a complete and exact definition of computation (Church-Turing thesis) (Burgin, 1987). However, the absolute nature of the Turing machine was questioned by contemporary research (Cooper, 2012) (Cooper & Leeuwen, 2013) and challenged by adopting a more general formal definition of algorithm (Burgin, 2005).
Nevertheless, in spite of all efforts, the conception of computation remains too vague and ambiguous. This vagueness of the foundations of computing has resulted in a variety of approaches, including approaches that contradict each other. Abramsky summarizes the process of successive change of models of computation and their future perspectives as follows:
“Traditionally, the dynamics of computing systems, their unfolding behavior in space and time has been a mere means to the end of computing the function which specifies the algorithmic problem which the system is solving. In much of contemporary computing, the situation is reversed: the purpose of the computing system is to exhibit certain behaviour. (…) We need a theory of the dynamics of informatic processes, of interaction, and information flow, as a basis for answering such fundamental questions as: What is computed? What is a process? What are the analogues to Turing completeness and universality when we are concerned with processes and their behaviours, rather than the functions which they compute? (Abramsky, 2008)
Abramsky emphasizes that there is the need for second-generation models of computation, and in particular process models. The first generation models of computation originated from problems of formalization of mathematics and logic, while processes or agents, interaction, and information flow are results of recent developments of computers and computing. In the second-generation models of computation, previously isolated systems are replaced by processes and agents for which the interactions with each other and with the environment are fundamental. Hewitt too advocates an agent-type, Actor model of computation (Hewitt, 2012) which is suitable for modeling of physical (intrinsic) computation.
In the historical perspective, the development of the concept of computation on the practical level related to operations performed by people and physical objects used as computing devices, while on the theoretical level computation was represented by abstract models and processes.
Variety of current approaches to the concept of computation shows remarkable complexity that makes communication of related results and ideas increasingly difficult. We explicated present diversity of concepts and models in (Burgin & Dodig-Crnkovic, 2013) to highlight the necessity of establishing relationships and common understanding. The analysis of the present state of the art allowed us to discover basic structures inherent for computation and to develop a multifaceted typology of computations. We presented the structural framework of information processing and computation with triadic relationships between (information processing (computation), algorithm and device/agent); (data, context/environment and function/goal); (structure, physical and mental/cognitive); (program, device and data), etc. Those are combined to form action computation pyramid with ((data, device/agent, program) and information processing/computation). An effective methodology of our approach is to find essential features of computation with the goal to explicate its nature and to build adequate models for research and technology. Our conclusion is that different models of computation may have their specific uses and applications, and it is necessary to understand their mutual relationships and the assumptions under which they apply in order to be able to consistently use them.
We underline several topics of importance for the development of new understanding of computing and its role: natural computation and the relationship between the model and physical implementation, interactivity as fundamental for computational modeling of concurrent information processing systems such as living organisms and their networks, and the new developments in modeling needed to support this generalized framework.
In such a way, we achieve better understanding of computation as information processing than we had before. As there is no information without (physical) representation (Landauer, 1996), the dynamics of information in nature is implemented on different levels of granularity by different physical processes, including the level of computation performed by computing machines (designed computation), as well as by living organisms (intrinsic computation) (Dodig-Crnkovic, 2014).
There are still many open problems related to the nature of information and computation, as well as to their relationships. How is information dynamics implemented/represented in computational systems, in machines, as well as in living organisms? Are computers processing only data or information or can they be made to process knowledge as well? What do we know of computational processes in machines and living organisms and how these processes are related? What can we learn from natural computational processes that can be useful for the development of information systems and knowledge management?
Our aim is to contribute to the future development by the exposition and delimitation of possibilities of the unified concept of computation understood as information processing (information transformation).
References and Notes
Abramsky, S. (2008). Information, Processes and Games. In J. Benthem van & P. Adriaans (Eds.), Philosophy of Information (pp. 483–549). Amsterdam, The Netherlands: North Holland.
Burgin, M. (1987). The Notion of Algorithm and the Turing-Church Thesis. In Proc. of the VIII International Congress on Logic, Methodology and Philosophy of Science, v.5 part 1 (pp. 138–140).
Burgin, M. (2005). Super-Recursive Algorithms. New York: Springer-Verlag New York Inc.
Burgin, M., & Dodig-Crnkovic, G. (2011). Information and Computation – Omnipresent and Pervasive. In Information and Computation (pp. vii –xxxii). New York/London/Singapore: World Scientific Pub Co Inc.
Burgin, M., & Dodig-Crnkovic, G. (2013). Typologies of Computation and Computational Models. Arxiv.org, arXiv:1312.
Cooper, S. B. (2012). The Mathematician’s Bias - and the Return to Embodied Computation. In H. Zenil (Ed.), A Computable Universe: Understanding and Exploring Nature as Computation. World Scientific Pub Co Inc.
Cooper, S. B., & Leeuwen, J. van. (2013). Alan Turing. His work and impact. Elsevier Science.
Copeland, B. J. (1996). What is computation? Synthese, 108(3), 335–359.
Denning, P. (2010). What is computation?: Editor’s Introduction. Ubiquity, (October), 1–2.
Denning, P. (2014). Structure and Organization of Computing. In J. Tucker, A., Gonzalez, T., Topi, H. and Diaz-Herrera (Ed.), Computing Handbook. Computer Science and Software Engineering. Chapman & Hall/CRC.
Dodig-Crnkovic, G. (2011). Dynamics of Information as Natural Computation. Information, 2(3), 460–477.
Dodig-Crnkovic, G. (2014). Modeling Life as Cognitive Info-Computation. In A. Beckmann, E. Csuhaj-Varjú, & K. Meer (Eds.), Computability in Europe 2014. LNCS (pp. 153–162). Berlin Heidelberg: Springer.
Dodig-Crnkovic, G., & Giovagnoli, R. (2013). Computing Nature. Berlin Heidelberg: Springer.
Hewitt, C. (2012). What is computation? Actor Model versus Turing’s Model. In H. Zenil (Ed.), A Computable Universe, Understanding Computation & Exploring Nature As Computation. World Scientific Publishing Company/Imperial College Press.
Kolmogorov, A. N. (1953). On the Concept of Algorithm. Russian Mathematical Surveys, 8(4), 175–176.
Landauer, R. (1996). The Physical Nature of Information. Physics Letter A, 217, 188.
Turing, A. M. (1936). On computable numbers, with an application to the Entscheidungs problem. Proceedings of the London Mathematical Society, 42(42), 230–265. doi:10.1112/plms/s2-42.1.23
Zenil, H. (2012). A Computable Universe. Understanding Computation & Exploring Nature As Computation. (H. Zenil, Ed.). Singapore: World Scientific Publishing Company/Imperial College Press.
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Introduction
Research in IT security often comes with decisions and possibilities that may or may not be considered ethical. However, it is often hard for young researchers to estimate the impact of their work, possible consequences and overall morality, as well as to where to draw the line. In some cases it is likely that more than hundreds of thousands of users will be affected, and it is unclear what is in their best interest: removal of a threat? Or rather a deeper analysis of the threat, which could prevent further vulnerabilities or attacks that are similar? In most cases, this decision is then left to the advisor who may have conflicting interests.
Methods
In this talk we will present recent borderline papers from the ethical point of view, and their implications on users. These papers are either directly or indirectly related to our own work, meaning that we will put our own perceptions on morality and ethics in perspective. We will furthermore present fundamental ethical principles which should not only be considered for IT security research, but can be applied to research in general. We argue that the establishment of ethical guidelines or frameworks without prior discussion and consensus in the research community probably would not lead to clarity on which lines in academic research should not be crossed. Especially the world-wide context of IT research poses challenges which are not easy to overcome: while researchers at US institutions are often forced to go through an IRB approval, nothing comparable exists in the broader context of European research. On the other hand, Europe with its much stronger privacy laws has nothing comparable within the US or Asia.
A good example is the analysis of a botnet [1]: malicious software which is run on thousands of computers, operated by unsuspecting users. These computers are then used for sending spam e-mails, collecting banking information, and many more severe malicious activities. The researcher is now in the dilemma: shut down the botnet which is just one among many more, or conducting a deeper analysis of the botnet? Another option would be to sanitize the computer and fix the underlying vulnerabilities of the computer, to prevent similar and future infections. Or should the user be warned, and made aware of the fact that the system is running malware including instructions on how to get rid of it?
The interest of the user, in particular for experiments where the user is not informed (or asked for consensus), has to be the highest priority. Another priority should be in our opinion that watching users getting harmed is not acceptable. Watching how their personal information is stolen or their computers are being abused for sending spam e-mails is not acceptable. But where to draw the line? What is acceptable, and what isn’t? And how can unethical research be punished, when the decision for publication is made by very few, namely the chair of a program committee or the editor of a journal?
Another good example is the Tor network [2], an anonymity network used by thousands of people on a daily basis to stay anonymous on the Internet. Due to the open design of the Tor network and the possibility that everyone can become part of the network and relay traffic for users, it can be tempting to modify traffic in the users best interest, e.g. by blocking malicious domains, inspecting and modifying file transfers or attacking the Tor network by trying to deanonymize its users. But how can deanonymization attacks be evaluated without assessing the impact on real users? In particular for complex systems, simulation is not always possible, and while there are simulation frameworks available for Tor, how can the researcher be sure that all important parameters of the simulation are correct and within expectation?
Conclusions
Neither researchers nor research subjects, e.g. malware authors, users or online services are currently in a position where they can refer to ethical principles that are accepted in the research community as well as on a larger scale. This talk is hopefully another step towards a guideline which is acceptable for all parties. However, much more research is needed in that direction, and the different interests of all stakeholders have to be balanced.
Acknowledgments
This talk extends prior work by the authors published in 2013 [3].
References and Notes
- Stone-Gross, Brett, et al. “Your botnet is my botnet: analysis of a botnet takeover.” Proceedings of the 16th ACM conference on Computer and communications security. ACM, 2009.
- Dingledine, Mathewson, et al. “Tor: The Second-generation Onion Router” Proceedings of the 13th Conference on USENIX Security Symposium, 2004
- Schrittwieser, Sebastian, Martin Mulazzani, and Edgar Weippl. “Ethics in security research which lines should not be crossed?” Security and Privacy Workshops (SPW), 2013 IEEE. IEEE, 2013.
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The standard model of evolution assumes a fixed fitness landscape. Usually there is co-evolution, though: besides being influenced by its environment, an agent also shapes its environment (as described by niche construction (Laland, Odling-Smee, & Feldman, 2001)). View this as a swamp-like fitness landscape that changes as an agent moves through it and acts in it.
This interplay is described on different similar aspects: between 'natural and cultural', 'social and infrastructure', 'function and structure', 'society and technology', 'decisions and acts', 'theory and practice' and 'micro and macro'. This mechanism can be used to explain certain dynamics. In the next paragraphs I will do so for technology and democracy. In general, out of the interactions of local elements, there is a bigger structure that emerges. This structure could then impose itself onto the agents, so that a status quo is reached: agents are influenced by the structure, while they don't have any more influence in return (Stirner, 1995, Stewart, 2014).
Technology is in interaction with a certain kind of society and ideas. Technology strengthens a certain type of society, while it is also out of current ideas that a technology is created. Technology creates the circumstances, the environment, in which one can act. We can find support in technology to liberate ourselves. But technology can never liberate in itself, because you can only liberate yourself. Technology can't save us, because then we aren't the drivers, the players, of our own future. Technology can reinforce certain liberating tendencies, but if these tendencies aren't present, even the most liberating technology will evolve to serve the current system.
Today’s democracy creates a sharp separation between decision making and acting. Some politicians make the decisions, which other people put into practice. This makes it possible to avoid responsibility, and creates alienation. Dreams can't evolve into acts.
Distributed governance is a step in the right direction. But often there is the assumption that we should make a global decision, and then all act by that decision, for example in (Banathy, 2000). Although these decisions and acts have come about in a distributed way, there is still a separation between them. A global decision is made out of local decisions, which lead to local acts bringing forth a global act. Another practice is where local decisions lead to local acts, out of which a global behavior, a global direction, emerges.
A solution to this structure that imposes itself could be a more hybrid structure, one that is constantly evolving, a variation and selection of different ways of organizing (Veitas & Weinbaum, 2014). There is not one utility measure that imposes a hierarchical ordering (Roughgarden, 2013). Instead of trying to reach a global, united decision or view, there would be local groups or individuals who develop themselves and work together to do so. It would be diverse and even contradictory. This conflict will boost a dynamic play.
Only a constant opposition can work, though, against the natural tendency of a system for unification, for getting stuck in a status quo. This mechanism is analogous to the second law of thermodynamics, which states that without selection, a system will become more and more disordered. Without opposition, a system will impose itself and become rigid. But this doesn't mean that a complex or anti-authoritarian society is impossible.
The idea is to create the environment that helps people to develop and enables them. But they are two different perspectives to do this (Busseniers, 2014): to start from yourself, constructing the world you would like to live in, or to start from the other, constructing a world where an assumed better behavior is more easily achieved. This last perspective is that of libertarian paternalism (Heylighen, 2009).
These considerations are important to take into account when thinking about the global brain. Will the global brain be this integrating structure, a stable attractor state, impossible to resist since it is omnipotent and omnipresent (Heylighen, 2014)? Or will it be a constantly evolving structure that enables us to build the world we want? Will it alienate our decisions from our acts? People could get stuck in a virtual world where they can raise all kinds of opinions, but without these being connected to their acts and everyday lives. But the internet could also enable people to put their ideas into practice, by providing tools, resources and people. Consider for example the scientific process. Right now, a researcher develops a plan for an experiment, performs an experiment and writes down the results in an article, and only then his ideas are peer-reviewed. At that stage, they might find out that actually there are some problems with the experimental setup. A more continuous peer-review could be interesting, where every step gets peer-reviewed. The global brain could enable this.
References
- Banathy, B. H. (2000). Guided evolution of society: A systems view. Kluwer Academic/Plenum Publishers New York.
- Busseniers E. (2014): Is external control important for internal control? (GBI Working paper)
- Heylighen, F. (2009). Stimuleren van geluk en sociale vooruitgang: een libertair paternalistische benadering. Ethiek en Maatschappij, 12(1), 147–167.
- Heylighen, F. (2014). Return to Eden? Promises and Perils on the Road to a Global Superintelligence. The End of the Beginning: Life, Society and Economy on the Brink of the Singularity.
- Laland, K. N., Odling-Smee, J., & Feldman, M. W. (2001). Cultural niche construction and human evolution. Journal of Evolutionary Biology, 14(1), 22–33. doi:10.1046/j.1420-9101.2001.00262.x
- Roughgarden, J. (2013). Evolution’s rainbow: Diversity, gender, and sexuality in nature and people. Univ of California Press.
- Stewart, J. E. (2014). The direction of evolution: The rise of cooperative organization. Biosystems, 123, 27–36. doi:10.1016/j.biosystems.2014.05.006
- Stirner, M. (1995). Stirner: The Ego and Its Own. Cambridge University Press.
- Veitas, V., & Weinbaum, D. (2014). A World of Views: A World of Interacting Post-human Intelligences. arXiv preprint arXiv:1410.6915.
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