Visualizing Discrete and Continuous Structures: Daisy Trees, Phase-State Surfaces, Exotic Loops, and Curvature Anomalies A Computational Study with Mathematical Guarantees

Visualizing Discrete and Continuous Structures: Daisy Trees, Phase-State Surfaces, Exotic Loops, and Curvature Anomalies A Computational Study with Mathematical Guarantees | Authorea try { document.documentElement.classList.add('js'); } catch (e) { } var _gaq = _gaq || []; _gaq.push(['_setAccount', 'G-8VDV14Y67G']); _gaq.push(['_trackPageview']); (function() { var ga = document.createElement('script'); ga.type = 'text/javascript'; ga.async = true; ga.src = ('https:' == document.location.protocol ? 'https://ssl' : 'http://www') + '.google-analytics.com/ga.js'; var s = document.getElementsByTagName('script')[0]; s.parentNode.insertBefore(ga, s); })(); Skip to main content Preprints Collections Wiley Open Research IET Open Research Ecological Society of Japan All Collections About About Authorea FAQs Contact Us Quick Search anywhere Search for preprint articles, keywords, etc. Search Search ADVANCED SEARCH SCROLL This is a preprint and has not been peer reviewed. Data may be preliminary. 22 September 2025 V1 Latest version Share on Visualizing Discrete and Continuous Structures: Daisy Trees, Phase-State Surfaces, Exotic Loops, and Curvature Anomalies A Computational Study with Mathematical Guarantees Author : Parker Emmerson 0009-0007-1288-3292 [email protected] Authors Info & Affiliations https://doi.org/10.22541/au.175856970.06483404/v1 156 views 103 downloads Contents Abstract Supplementary Material Information & Authors Metrics & Citations View Options References Figures Tables Media Share Abstract We present and analyze a Python program that visualizes four mathematical constructions: (i) a q-regular rooted "daisy tree"; (ii) a truncated phase-state function H τ over the complex plane; (iii) a non-holomorphic iterative system that generates fractal "exotic loops"; and (iv) a curvature field with angular modulation and a divergence-free flow. We provide precise definitions, structural properties, and convergence conditions in theorems, lemmas, and propositions. We also include guidance and placeholders for figures rendered by the program and the complete source code verbatim for reproducibility. Contents Supplementary Material File (visualizing_discrete_and_continuous_structures__daisy_trees__phase_state_surfaces__exotic_loops__and_curvature_anomalies (1).pdf) Download 1.64 MB Information & Authors Information Version history V1 Version 1 22 September 2025 Copyright This work is licensed under a Creative Commons Attribution 4.0 International License Keywords complex plane. curvature daisy networks exotic loops trees Authors Affiliations Parker Emmerson 0009-0007-1288-3292 [email protected] View all articles by this author Metrics & Citations Metrics Article Usage 156 views 103 downloads .FvxKWukQNSOunydq8rnd { width: 100px; } Citations Download citation Parker Emmerson. Visualizing Discrete and Continuous Structures: Daisy Trees, Phase-State Surfaces, Exotic Loops, and Curvature Anomalies A Computational Study with Mathematical Guarantees. Authorea . 22 September 2025. DOI: https://doi.org/10.22541/au.175856970.06483404/v1 If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download. For more information or tips please see 'Downloading to a citation manager' in the Help menu . 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