In this thesis we will study conditions for the existence of minimal sized omnipatterns in higher dimensions. We will introduce recent work conducted on one dimensional and two dimensional patterns known as omnisequences and omnimosaics, respectively. These have been studied by Abraham et al [3] and Banks et al [2]. The three dimensional patterns we study are called omnisculptures, and will be the focus of this thesis. A (K,a) omnisequence of length n is a string of letters that contains each of the ak words of length k over [A]={1,2,...a} as a substring. An omnimosaic O(n,k,a) is an n × n matrix, with entries from the set A ={1,2,...,a}, that contains each of the {ak2} k × k matrices over A as a submatrix. An omnisculpture is an n × n × n sculpture (a three dimensional matrix) with entries from set A ={1,2,...,a} that contains all the ak3 k × k × k subsculptures as an embedded submatrix of the larger sculpture. We will show that for given k, the existence of a minimal omnisculpture is guaranteed when kak2/3/e ≤ n ≤kak2/3/e(1+ε) and ε=εk → 0 is a sufficiently small function of k.
","abstract_html":"<p>In this thesis we will study conditions for the existence of minimal sized omnipatterns in higher dimensions. We will introduce recent work conducted on one dimensional and two dimensional patterns known as omnisequences and omnimosaics, respectively. These have been studied by Abraham et al [3] and Banks et al [2]. The three dimensional patterns we study are called omnisculptures, and will be the focus of this thesis. A (<em>K</em>,<em>a</em>) omnisequence of length <em>n</em> is a string of letters that contains each of the <em>a<sup>k</sup></em> words of length <em>k</em> over [<em>A</em>]={1,2,...a} as a substring. An omnimosaic <em>O</em>(<em>n</em>,<em>k</em>,<em>a</em>) is an <em>n</em> × <em>n</em> matrix, with entries from the set <em>A</em> ={1,2,...,<em>a</em>}, that contains each of the {<em>a<sup>k<sup>2</sup></sup></em>} <em>k</em> × <em>k</em> matrices over <em>A</em> as a submatrix. An omnisculpture is an <em>n</em> × <em>n</em> × <em>n</em> sculpture (a three dimensional matrix) with entries from set <em>A</em> ={1,2,...,<em>a</em>} that contains all the <em>a<sup>k<sup>3</sup></sup></em> <em>k</em> × <em>k</em> × <em>k</em> subsculptures as an embedded submatrix of the larger sculpture. We will show that for given <em>k</em>, the existence of a minimal omnisculpture is guaranteed when <em>ka<sup>k<sup>2</sup>/3</sup>/e ≤ n ≤ka<sup>k<sup>2</sup>/3</sup>/e</em>(1+ε) and <em>ε=ε<sub>k</sub></em> → 0 is a sufficiently small function of <em>k</em>.</p>","abstract_has_math":false,"creators":["Eroglu, Cihan"],"institution":null,"degree_name":"MS (Master of Science)","degree_level":"Thesis - unrestricted","degree_discipline":"Mathematical Sciences","degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-08-17T07:00:00Z","date_published":"2011-08-17T07:00:00Z","updated_at":"2026-07-24T02:20:14Z","subjects":["Omnisculptures","Omnimosaics","Graphs","Omnisequences","Discrete Mathematics and Combinatorics","Mathematics","Physical Sciences and Mathematics"],"languages":[],"rights":["Copyright by the authors."],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://dc.etsu.edu/etd/1346","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Eroglu, Cihan"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.available","label":"Dc Date Available","values":["1990-01-01T08:00:00Z"]},{"key":"dc:date.issued","label":"Date","values":["2011-08-17T07:00:00Z"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematical Sciences"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Thesis - unrestricted"]},{"key":"thesis:degree_name","label":"Degree Name","values":["MS (Master of Science)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Omnisculptures","Omnimosaics","Graphs","Omnisequences","Discrete Mathematics and Combinatorics","Mathematics","Physical Sciences and Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:rights","label":"Dc Rights","values":["Copyright by the authors."]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://dc.etsu.edu/context/etd/article/2537/viewcontent/ErogluC072411f.pdf","https://dc.etsu.edu/etd/1346"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["In this thesis we will study conditions for the existence of minimal sized omnipatterns in higher dimensions. We will introduce recent work conducted on one dimensional and two dimensional patterns known as omnisequences and omnimosaics, respectively. These have been studied by Abraham et al [3] and Banks et al [2]. The three dimensional patterns we study are called omnisculptures, and will be the focus of this thesis. A (K,a) omnisequence of length n is a string of letters that contains each of the ak words of length k over [A]={1,2,...a} as a substring. An omnimosaic O(n,k,a) is an n × n matrix, with entries from the set A ={1,2,...,a}, that contains each of the {ak2} k × k matrices over A as a submatrix. An omnisculpture is an n × n × n sculpture (a three dimensional matrix) with entries from set A ={1,2,...,a} that contains all the ak3 k × k × k subsculptures as an embedded submatrix of the larger sculpture. We will show that for given k, the existence of a minimal omnisculpture is guaranteed when kak2/3/e ≤ n ≤kak2/3/e(1+ε) and ε=εk → 0 is a sufficiently small function of k.
"]},{"key":"dc:title","label":"Title","values":["Omnisculptures."]}]}],"canonical_facts":{"dc:creator":["Eroglu, Cihan"],"dc:date.available":["1990-01-01T08:00:00Z"],"dc:date.issued":["2011-08-17T07:00:00Z"],"dc:description.abstract":["In this thesis we will study conditions for the existence of minimal sized omnipatterns in higher dimensions. We will introduce recent work conducted on one dimensional and two dimensional patterns known as omnisequences and omnimosaics, respectively. These have been studied by Abraham et al [3] and Banks et al [2]. The three dimensional patterns we study are called omnisculptures, and will be the focus of this thesis. A (K,a) omnisequence of length n is a string of letters that contains each of the ak words of length k over [A]={1,2,...a} as a substring. An omnimosaic O(n,k,a) is an n × n matrix, with entries from the set A ={1,2,...,a}, that contains each of the {ak2} k × k matrices over A as a submatrix. An omnisculpture is an n × n × n sculpture (a three dimensional matrix) with entries from set A ={1,2,...,a} that contains all the ak3 k × k × k subsculptures as an embedded submatrix of the larger sculpture. We will show that for given k, the existence of a minimal omnisculpture is guaranteed when kak2/3/e ≤ n ≤kak2/3/e(1+ε) and ε=εk → 0 is a sufficiently small function of k.
"],"dc:identifier":["https://dc.etsu.edu/context/etd/article/2537/viewcontent/ErogluC072411f.pdf","https://dc.etsu.edu/etd/1346"],"dc:rights":["Copyright by the authors."],"dc:subject":["Omnisculptures","Omnimosaics","Graphs","Omnisequences","Discrete Mathematics and Combinatorics","Mathematics","Physical Sciences and Mathematics"],"dc:title":["Omnisculptures."],"thesis:degree_discipline":["Mathematical Sciences"],"thesis:degree_level":["Thesis - unrestricted"],"thesis:degree_name":["MS (Master of Science)"]},"updated_at":"2026-07-24T02:20:14Z"}