Topology generated non-fungible tokens: blockchain as infrastructure for a circular economy in architectural design.

The paper presents a new digital infrastructure layer for buildings and architectural assets. The infrastructure layer consists of a combination of topology graphs secured on a decentralised ledger. The topology graphs organise non-fungible digital tokens which each represent and correspond to building components, and in the root of the graph to the building itself. The paper presents background research in the relationship of building representation in the form of graphs with topology, of both manifold and non manifold nature. In parallel we present and analyse the relationship between digital representation and physical manifestation of a building, and back again. Within the digital representations the paper analyses the securing and saving of information on decentralised ledger technologies (such as blockchain). We then present a simple sample of generating and registering a non-manifold topology graph on the Ethereum blockchain as an EC721 token, i.e. a digital object that is unique, all through the use of dynamo and python scripting connected with a smart contract on the Ethereum blockchain. Ownership of this token can then be transferred on the blockchain smart contracts. The paper concludes with a discussion of the possibilities that this integration brings in terms of material passports and a circular economy and smart contracts as an infrastructure for whole-lifecycle BIM and digitally encapsulates of value in architectural design

Fixed point sets in digital topology, 2

[EN] We continue the work of [10], studying properties of digital images determined by fixed point invariants. We introduce pointed versions of invariants that were introduced in [10]. We introduce freezing sets and cold sets to show how the existence of a fixed point set for a continuous self-map restricts the map on the complement of the fixed point set.Boxer, L. (2020). Fixed point sets in digital topology, 2. Applied General Topology. 21(1):111-133. https://doi.org/10.4995/agt.2020.12101OJS111133211C. Berge, Graphs and Hypergraphs, 2nd edition, North-Holland, Amsterdam, 1976.L. Boxer, Digitally Continuous functions, Pattern Recognition Letters 15 (1994), 833-839. https://doi.org/10.1016/0167-8655(94)90012-4L. Boxer, A classical construction for the digital fundamental group, Journal of Mathematical Imaging and Vision 10 (1999), 51-62. https://doi.org/10.1023/A:1008370600456L. Boxer, Generalized normal product adjacency in digital topology, Applied General Topology 18, no. 2 (2017), 401-427. https://doi.org/10.4995/agt.2017.7798L. Boxer, Alternate product adjacencies in digital topology, Applied General Topology 19, no. 1 (2018), 21-53. https://doi.org/10.4995/agt.2018.7146L. Boxer, Fixed points and freezing sets in digital topology, Proceedings, Interdisciplinary Colloquium in Topology and its Applications in Vigo, Spain; 55-61.L. Boxer, O. Ege, I. Karaca, J. Lopez and J. Louwsma, Digital fixed points, approximate fixed points, and universal functions, Applied General Topology 17, no. 2 (2016), 159-172. https://doi.org/10.4995/agt.2016.4704L. Boxer and I. Karaca, Fundamental groups for digital products, Advances and Applications in Mathematical Sciences 11, no. 4 (2012), 161-180.L. Boxer and P. C. Staecker, Fundamental groups and Euler characteristics of sphere-like digital images, Applied General Topology 17, no. 2 (2016), 139-158. https://doi.org/10.4995/agt.2016.4624L. Boxer and P. C. Staecker, Fixed point sets in digital topology, 1, Applied General Topology, to appear.G. Chartrand and L. Lesniak, Graphs & Digraphs, 2nd ed., Wadsworth, Inc., Belmont, CA, 1986.J. Haarmann, M. P. Murphy, C. S. Peters and P. C. Staecker, Homotopy equivalence in finite digital images, Journal of Mathematical Imaging and Vision 53 (2015), 288-302. https://doi.org/10.1007/s10851-015-0578-8S.-E. Han, Non-product property of the digital fundamental group, Information Sciences 171 (2005), 73-91. https://doi.org/10.1016/j.ins.2004.03.018E. Khalimsky, Motion, deformation, and homotopy in finite spaces, in Proceedings IEEE Intl. Conf. on Systems, Man, and Cybernetics, 1987, 227-234.A. Rosenfeld, Digital topology, The American Mathematical Monthly 86, no. 8 (1979), 621-630. https://doi.org/10.1080/00029890.1979.11994873A. Rosenfeld, 'Continuous' functions on digital pictures, Pattern Recognition Letters 4 (1986), 177-184. https://doi.org/10.1016/0167-8655(86)90017-

Topology Generated Non-Fungible Tokens - Blockchain as infrastructure for a circular economy in architectural design

The paper presents a new digital infrastructure layer for buildings and architectural assets. The infrastructure layer consists of a combination of topology graphs secured on a decentralised ledger. The topology graphs organise non-fungible digital tokens which each represent and correspond to building components, and in the root of the graph to the building itself.The paper presents background research in the relationship of building representation in the form of graphs with topology, of both manifold and non manifold nature. In parallel we present and analyse the relationship between digital representation and physical manifestation of a building, and back again. Within the digital representations the paper analyses the securing and saving of information on decentralised ledger technologies (such as blockchain). We then present a simple sample of generating and registering a non-manifold topology graph on the Ethereum blockchain as an EC721 token, i.e. a digital object that is unique, all through the use of dynamo and python scripting connected with a smart contract on the Ethereum blockchain. Ownership of this token can then be transferred on the blockchain smart contracts. The paper concludes with a discussion of the possibilities that this integration brings in terms of material passports and a circular economy and smart contracts as an infrastructure for whole-lifecycle BIM and digitally encapsulates of value in architectural designPlease write your abstract here by clicking this paragraph

Fixed poin sets in digital topology, 1

[EN] In this paper, we examine some properties of the fixed point set of a digitally continuous function. The digital setting requires new methods that are not analogous to those of classical topological fixed point theory, and we obtain results that often differ greatly from standard results in classical topology. We introduce several measures related to fixed points for continuous self-maps on digital images, and study their properties. Perhaps the most important of these is the fixed point spectrum F(X) of a digital image: that is, the set of all numbers that can appear as the number of fixed points for some continuous self-map. We give a complete computation of F(Cn) where Cn is the digital cycle of n points. For other digital images, we show that, if X has at least 4 points, then F(X) always contains the numbers 0, 1, 2, 3, and the cardinality of X. We give several examples, including Cn, in which F(X) does not equal {0, 1, . . . , #X}. We examine how fixed point sets are affected by rigidity, retraction, deformation retraction, and the formation of wedges and Cartesian products. We also study how fixed point sets in digital images can be arranged; e.g., for some digital images the fixed point set is always connected.Boxer, L.; Staecker, PC. (2020). Fixed poin sets in digital topology, 1. Applied General Topology. 21(1):87-110. https://doi.org/10.4995/agt.2020.12091OJS87110211C. Berge, Graphs and Hypergraphs, 2nd edition, North-Holland, Amsterdam, 1976.L. Boxer, Digitally continuous functions, Pattern Recognition Letters 15 (1994), 833-839. https://doi.org/10.1016/0167-8655(94)90012-4L. Boxer, A classical construction for the digital fundamental group, Journal of Mathematical Imaging and Vision 10 (1999), 51-62. https://doi.org/10.1023/A:1008370600456L. Boxer, Continuous maps on digital simple closed curves, Applied Mathematics 1 (2010), 377-386. https://doi.org/10.4236/am.2010.15050L. Boxer, Generalized normal product adjacency in digital topology, Applied General Topology 18, no. 2 (2017), 401-427. https://doi.org/10.4995/agt.2017.7798L. Boxer, Alternate product adjacencies in digital topology, Applied General Topology 19, no. 1 (2018), 21-53. https://doi.org/10.4995/agt.2018.7146L. Boxer, Fixed points and freezing sets in digital topology, Proceedings, 2019 Interdisciplinary Colloquium in Topology and its Applications, in Vigo, Spain; 55-61.L. Boxer, O. Ege, I. Karaca, J. Lopez, and J. Louwsma, Digital fixed points, approximate fixed points, and universal functions, Applied General Topology 17, no. 2 (2016), 159-172. https://doi.org/10.4995/agt.2016.4704L. Boxer and I. Karaca, Fundamental groups for digital products, Advances and Applications in Mathematical Sciences 11, no. 4 (2012), 161-180.L. Boxer and P. C. Staecker, Remarks on fixed point assertions in digital topology, Applied General Topology 20, no. 1 (2019), 135-153. https://doi.org/10.4995/agt.2019.10474J. Haarmann, M. P. Murphy, C. S. Peters and P. C. Staecker, Homotopy equivalence in finite digital images, Journal of Mathematical Imaging and Vision 53 (2015), 288-302. https://doi.org/10.1007/s10851-015-0578-8B. Jiang, Lectures on Nielsen fixed point theory, Contemporary Mathematics 18 (1983). https://doi.org/10.1090/conm/014E. Khalimsky, Motion, deformation, and homotopy in finite spaces, in Proceedings IEEE Intl. Conf. on Systems, Man, and Cybernetics (1987), 227-234.A. Rosenfeld, "Continuous" functions on digital pictures, Pattern Recognition Letters 4 (1986), 177-184. https://doi.org/10.1016/0167-8655(86)90017-6P. C. Staecker, Some enumerations of binary digital images, arXiv:1502.06236, 2015