Elementary Fractal Geometry. 2. Carpets Involving Irrational Rotations

Self-similar sets with the open set condition, the linear objects of fractal geometry, have been considered mainly for crystallographic data. Here we introduce new symmetry classes in the plane, based on rotation by irrational angles. Examples without characteristic directions, with strong connected...

Full description

Saved in:
Bibliographic Details
Published in:Fractal and fractional 2022-01, Vol.6 (1), p.39
Main Authors: Bandt, Christoph, Mekhontsev, Dmitry
Format: Article
Language:English
Subjects:
Algebra
Angles (geometry)
aperiodic tile
Carpets
Crystallography
fractal
Fractal geometry
Fractals
Geometry
Number theory
Porous materials
quadratic number field
self-similar
Self-similarity
Symmetry
Citations: Items that this one cites
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
cited_by
cites cdi_FETCH-LOGICAL-c338t-f266de29838536290ea95f9fdc14f6aa5112b87b529cb179a15760590390bf93
container_end_page
container_issue 1
container_start_page 39
container_title Fractal and fractional
container_volume 6
creator Bandt, Christoph
Mekhontsev, Dmitry
description Self-similar sets with the open set condition, the linear objects of fractal geometry, have been considered mainly for crystallographic data. Here we introduce new symmetry classes in the plane, based on rotation by irrational angles. Examples without characteristic directions, with strong connectedness and small complexity, were found in a computer-assisted search. They are surprising since the rotations are given by rational matrices, and the proof of the open set condition usually requires integer data. We develop a classification of self-similar sets by symmetry class and algebraic numbers. Examples are given for various quadratic number fields.
doi_str_mv 10.3390/fractalfract6010039
format article
fullrecord <record><control><sourceid>proquest_doaj_</sourceid><recordid>TN_cdi_doaj_primary_oai_doaj_org_article_c1915f2f7a3444fa9286e3af03cbdf0b</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><doaj_id>oai_doaj_org_article_c1915f2f7a3444fa9286e3af03cbdf0b</doaj_id><sourcerecordid>2621280502</sourcerecordid><originalsourceid>FETCH-LOGICAL-c338t-f266de29838536290ea95f9fdc14f6aa5112b87b529cb179a15760590390bf93</originalsourceid><addsrcrecordid>eNptkEFLAzEQhYMoWGp_gZcFz62TZJPdHKW0dUEQpPcwm03Klu2mJmmh_961K-LB03sMj29mHiGPFBacK3h2AU3C7ioSKABXN2TCBORzTinc_vH3ZBbjHgBYobiAYkLWq84ebJ8wXLL1CMo21h9sCpdFxhbZEsPRpphV_dl357bfZVUImFrfD8kPn642PpA7h120sx-dku16tV2-zt_eN9Xy5W1uOC_T3DEpG8tUyUvBJVNgUQmnXGNo7iSioJTVZVELpkxNC4VUFBKEGl6C2ik-JdWIbTzu9TG0h-Fu7bHV14EPO40htaaz2lBFhWOuQJ7nuUPFSmk5OuCmbhzUA-tpZB2D_zzZmPTen8LwVdRMMspKEMCGFB9TJvgYg3W_Wyno7_r1P_XzL690egE</addsrcrecordid><sourcetype>Open Website</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>2621280502</pqid></control><display><type>article</type><title>Elementary Fractal Geometry. 2. Carpets Involving Irrational Rotations</title><source>Publicly Available Content Database</source><creator>Bandt, Christoph ; Mekhontsev, Dmitry</creator><creatorcontrib>Bandt, Christoph ; Mekhontsev, Dmitry</creatorcontrib><description>Self-similar sets with the open set condition, the linear objects of fractal geometry, have been considered mainly for crystallographic data. Here we introduce new symmetry classes in the plane, based on rotation by irrational angles. Examples without characteristic directions, with strong connectedness and small complexity, were found in a computer-assisted search. They are surprising since the rotations are given by rational matrices, and the proof of the open set condition usually requires integer data. We develop a classification of self-similar sets by symmetry class and algebraic numbers. Examples are given for various quadratic number fields.</description><identifier>ISSN: 2504-3110</identifier><identifier>EISSN: 2504-3110</identifier><identifier>DOI: 10.3390/fractalfract6010039</identifier><language>eng</language><publisher>Basel: MDPI AG</publisher><subject>Algebra ; Angles (geometry) ; aperiodic tile ; Carpets ; Crystallography ; fractal ; Fractal geometry ; Fractals ; Geometry ; Number theory ; Porous materials ; quadratic number field ; self-similar ; Self-similarity ; Symmetry</subject><ispartof>Fractal and fractional, 2022-01, Vol.6 (1), p.39</ispartof><rights>2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License.</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-c338t-f266de29838536290ea95f9fdc14f6aa5112b87b529cb179a15760590390bf93</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://www.proquest.com/docview/2621280502/fulltextPDF?pq-origsite=primo$$EPDF$$P50$$Gproquest$$Hfree_for_read</linktopdf><linktohtml>$$Uhttps://www.proquest.com/docview/2621280502?pq-origsite=primo$$EHTML$$P50$$Gproquest$$Hfree_for_read</linktohtml><link.rule.ids>314,776,780,25732,27903,27904,36991,44569,74872</link.rule.ids></links><search><creatorcontrib>Bandt, Christoph</creatorcontrib><creatorcontrib>Mekhontsev, Dmitry</creatorcontrib><title>Elementary Fractal Geometry. 2. Carpets Involving Irrational Rotations</title><title>Fractal and fractional</title><description>Self-similar sets with the open set condition, the linear objects of fractal geometry, have been considered mainly for crystallographic data. Here we introduce new symmetry classes in the plane, based on rotation by irrational angles. Examples without characteristic directions, with strong connectedness and small complexity, were found in a computer-assisted search. They are surprising since the rotations are given by rational matrices, and the proof of the open set condition usually requires integer data. We develop a classification of self-similar sets by symmetry class and algebraic numbers. Examples are given for various quadratic number fields.</description><subject>Algebra</subject><subject>Angles (geometry)</subject><subject>aperiodic tile</subject><subject>Carpets</subject><subject>Crystallography</subject><subject>fractal</subject><subject>Fractal geometry</subject><subject>Fractals</subject><subject>Geometry</subject><subject>Number theory</subject><subject>Porous materials</subject><subject>quadratic number field</subject><subject>self-similar</subject><subject>Self-similarity</subject><subject>Symmetry</subject><issn>2504-3110</issn><issn>2504-3110</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2022</creationdate><recordtype>article</recordtype><sourceid>PIMPY</sourceid><sourceid>DOA</sourceid><recordid>eNptkEFLAzEQhYMoWGp_gZcFz62TZJPdHKW0dUEQpPcwm03Klu2mJmmh_961K-LB03sMj29mHiGPFBacK3h2AU3C7ioSKABXN2TCBORzTinc_vH3ZBbjHgBYobiAYkLWq84ebJ8wXLL1CMo21h9sCpdFxhbZEsPRpphV_dl357bfZVUImFrfD8kPn642PpA7h120sx-dku16tV2-zt_eN9Xy5W1uOC_T3DEpG8tUyUvBJVNgUQmnXGNo7iSioJTVZVELpkxNC4VUFBKEGl6C2ik-JdWIbTzu9TG0h-Fu7bHV14EPO40htaaz2lBFhWOuQJ7nuUPFSmk5OuCmbhzUA-tpZB2D_zzZmPTen8LwVdRMMspKEMCGFB9TJvgYg3W_Wyno7_r1P_XzL690egE</recordid><startdate>20220101</startdate><enddate>20220101</enddate><creator>Bandt, Christoph</creator><creator>Mekhontsev, Dmitry</creator><general>MDPI AG</general><scope>AAYXX</scope><scope>CITATION</scope><scope>8FE</scope><scope>8FG</scope><scope>ABJCF</scope><scope>ABUWG</scope><scope>AFKRA</scope><scope>AZQEC</scope><scope>BENPR</scope><scope>BGLVJ</scope><scope>CCPQU</scope><scope>DWQXO</scope><scope>HCIFZ</scope><scope>L6V</scope><scope>M7S</scope><scope>PIMPY</scope><scope>PQEST</scope><scope>PQQKQ</scope><scope>PQUKI</scope><scope>PRINS</scope><scope>PTHSS</scope><scope>DOA</scope></search><sort><creationdate>20220101</creationdate><title>Elementary Fractal Geometry. 2. Carpets Involving Irrational Rotations</title><author>Bandt, Christoph ; Mekhontsev, Dmitry</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c338t-f266de29838536290ea95f9fdc14f6aa5112b87b529cb179a15760590390bf93</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2022</creationdate><topic>Algebra</topic><topic>Angles (geometry)</topic><topic>aperiodic tile</topic><topic>Carpets</topic><topic>Crystallography</topic><topic>fractal</topic><topic>Fractal geometry</topic><topic>Fractals</topic><topic>Geometry</topic><topic>Number theory</topic><topic>Porous materials</topic><topic>quadratic number field</topic><topic>self-similar</topic><topic>Self-similarity</topic><topic>Symmetry</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Bandt, Christoph</creatorcontrib><creatorcontrib>Mekhontsev, Dmitry</creatorcontrib><collection>CrossRef</collection><collection>ProQuest SciTech Collection</collection><collection>ProQuest Technology Collection</collection><collection>Materials Science &amp; Engineering Collection</collection><collection>ProQuest Central (Alumni)</collection><collection>ProQuest Central</collection><collection>ProQuest Central Essentials</collection><collection>ProQuest Central</collection><collection>Technology Collection</collection><collection>ProQuest One Community College</collection><collection>ProQuest Central</collection><collection>SciTech Premium Collection</collection><collection>ProQuest Engineering Collection</collection><collection>Engineering Database</collection><collection>Publicly Available Content Database</collection><collection>ProQuest One Academic Eastern Edition (DO NOT USE)</collection><collection>ProQuest One Academic</collection><collection>ProQuest One Academic UKI Edition</collection><collection>ProQuest Central China</collection><collection>Engineering collection</collection><collection>DOAJ Directory of Open Access Journals</collection><jtitle>Fractal and fractional</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Bandt, Christoph</au><au>Mekhontsev, Dmitry</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Elementary Fractal Geometry. 2. Carpets Involving Irrational Rotations</atitle><jtitle>Fractal and fractional</jtitle><date>2022-01-01</date><risdate>2022</risdate><volume>6</volume><issue>1</issue><spage>39</spage><pages>39-</pages><issn>2504-3110</issn><eissn>2504-3110</eissn><abstract>Self-similar sets with the open set condition, the linear objects of fractal geometry, have been considered mainly for crystallographic data. Here we introduce new symmetry classes in the plane, based on rotation by irrational angles. Examples without characteristic directions, with strong connectedness and small complexity, were found in a computer-assisted search. They are surprising since the rotations are given by rational matrices, and the proof of the open set condition usually requires integer data. We develop a classification of self-similar sets by symmetry class and algebraic numbers. Examples are given for various quadratic number fields.</abstract><cop>Basel</cop><pub>MDPI AG</pub><doi>10.3390/fractalfract6010039</doi><oa>free_for_read</oa></addata></record>
fulltext fulltext
identifier ISSN: 2504-3110
ispartof Fractal and fractional, 2022-01, Vol.6 (1), p.39
issn 2504-3110
2504-3110
language eng
recordid cdi_doaj_primary_oai_doaj_org_article_c1915f2f7a3444fa9286e3af03cbdf0b
source Publicly Available Content Database
subjects Algebra
Angles (geometry)
aperiodic tile
Carpets
Crystallography
fractal
Fractal geometry
Fractals
Geometry
Number theory
Porous materials
quadratic number field
self-similar
Self-similarity
Symmetry
title Elementary Fractal Geometry. 2. Carpets Involving Irrational Rotations
url http://sfxeu10.hosted.exlibrisgroup.com/loughborough?ctx_ver=Z39.88-2004&ctx_enc=info:ofi/enc:UTF-8&ctx_tim=2025-01-23T11%3A22%3A09IST&url_ver=Z39.88-2004&url_ctx_fmt=infofi/fmt:kev:mtx:ctx&rfr_id=info:sid/primo.exlibrisgroup.com:primo3-Article-proquest_doaj_&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.genre=article&rft.atitle=Elementary%20Fractal%20Geometry.%202.%20Carpets%20Involving%20Irrational%20Rotations&rft.jtitle=Fractal%20and%20fractional&rft.au=Bandt,%20Christoph&rft.date=2022-01-01&rft.volume=6&rft.issue=1&rft.spage=39&rft.pages=39-&rft.issn=2504-3110&rft.eissn=2504-3110&rft_id=info:doi/10.3390/fractalfract6010039&rft_dat=%3Cproquest_doaj_%3E2621280502%3C/proquest_doaj_%3E%3Cgrp_id%3Ecdi_FETCH-LOGICAL-c338t-f266de29838536290ea95f9fdc14f6aa5112b87b529cb179a15760590390bf93%3C/grp_id%3E%3Coa%3E%3C/oa%3E%3Curl%3E%3C/url%3E&rft_id=info:oai/&rft_pqid=2621280502&rft_id=info:pmid/&rfr_iscdi=true