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Visitors can manipulate and visualize iterations used to construct the fractals. Geometric transformations—homothety, rotation, and translation—support HTML page hierarchy structures in both fractal modeling and virtual room construction. The approach supports classroom visualization via immersive glasses and can extend to augmented reality (AR) on smartphones or tablets, helping improve instruction in fractal geometry and related 3D graphic representation topics.",{"@graph":69,"@context":114},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/development-of-virtual-reality-environments-to-visualize-the-fractals-of-the-dragons-curve-using-platos-polyhedra/136510/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/development-of-virtual-reality-environments-to-visualize-the-fractals-of-the-dragons-curve-using-platos-polyhedra/136510.png","ImageObject",300,407,{"name":92,"@type":93},"Dozel","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-09-18","2026-08-22",true,{"@type":102,"interactionType":103,"userInteractionCount":34},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108],{"name":109,"@type":110,"acceptedAnswer":111},"Which geometric transformations are used during modeling and virtual room construction?","Question",{"text":112,"@type":113},"The work uses homothety, rotation, and translation transformations applied to HTML page hierarchy structures in both phases.","Answer","https://schema.org",{"og:url":83,"og:type":116,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":118,"canonical":83},"index,follow",{"doc_id":120,"site_id":62},136510,1787387987,{"code":4,"msg":5,"data":123},{"doc_id":120,"user_id":124,"nickname":92,"user_avatar":125,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":126,"file_id":127,"file_url":128,"file_type":129,"file_size":130,"view_count":34,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":131,"language":132,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":133,"faqs":134,"seo_title":135,"seo_description":67,"update_tm":121,"read_time":136},962085662650,"https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0","Development of Virtual Reality Environments to Visualize the Fractals of the Dragon’s Curve Using Plato’s Polyhedra  \nPaulo Henrique Siqueira 1  \n1 Graphic Expression Department, Federal University of Paraná, Curitiba, Brazil  \nCorrespondence: Paulo Henrique Siqueira, Graphic Expression Department, Federal University of Paraná, Curitiba, [P.O. box](P.O. box) 19081, Brazil.  \nReceived: August 10, 2024  \ndoi:10.5539/jel.v14n4p21  \nAccepted: October 21, 2024 Online Published: February 10, 2025 URL: [https://doi.org/10.5539/jel.v14n4p21](https://doi.org/10.5539/jel.v14n4p21)  \nAbstract  \nThis article presents the adaptations for creating a set of Dragon Curve fractals using the Platonic polyhedra. The modeled fractals were inserted into environments programmed with Virtual Reality (VR) resources, which allow the visitor to manipulate and visualize each iteration used to construct these fractals. The structures of HTML page hierarchies were used through geometric transformations of homothety, rotation and translation in both phases of this work: in the modeling of fractals and also in the construction of virtual rooms. The resources presented in this article can be used in the classroom to visualize polyhedra fractals using immersive glasses, in addition to Augmented Reality (AR) using smartphones or tablets. The objective of this article is to show the use of simple and free technologies, which can be used to create teaching materials with a great contribution to improving the teaching of Fractal Geometry, in addition to other areas that use graphic representations of 3D objects.  \nKeywords: virtual reality, fractals, Platonic polyhedra, dragon curve  \n1. Introduction  \nStudents’ understanding of subject content that involves concepts in three dimensions (3D) becomes more efficient and attractive with the use of auxiliary resources. Applications and web environments significantly contribute to the learning of Physics (Komikesari et al., 2019; Rozal, Ananda, Zb, Fauziddin, & Sulman, 2021) and Biology concepts (Delgado, Bhark & Donahue, 2021) and have been used as an interesting alternative to help of students learning. The concrete materials, made with 3D printers, can be used in Biology classes (Olumorin, Babalola, & Ayoola, 2022), Mathematics (Quigley, 2021) and in subjects with content that involves development and spatial skills (Karaismailoglu & Yildirim, 2023) .  \nLearning content that involves three-dimensional concepts can be more effective through the use of objects modeled with virtual technologies. Virtual Reality (VR) allows the creation of an immersive and interactive environment with the objects manipulation through controls and immersive glasses (Marougkas, Troussas, Krouska, & Sgouropoulou, 2023) .  \nThe creation of programmed environments in VR can help in constructions simulations (Soliman, Pesyridis, Dalaymani-Zad, Gronfula, & Kourmpetis, 2021), simulations of training situations (de Geus et al., 2020), Medicine (Pottle, 2019), educational games (Pirker, Dengel, Holly, & Safikhani, 2020), visualizations of physical, biological or chemical phenomena (Kumar et al., 2021), and other areas linked to education (Melinda & Widjaja, 2022) .  \nThe polyhedra used in Fractal Geometry are defined with the characteristics of 2D fractals, expanding the concepts used in 2D constructions to 3D objects (Jacomeli & Souza, 2020) . Studies on 3D fractals are expanding (Husain, Gomada, Megham, & Asish, 2021), and these solids help in studies of the applications of these objects in tree leaves and algae, blood and lung vessels, clouds and rain areas, coastal craters, mountain ranges, snowflakes, among others (Nutu & Axinte, 2022) .  \nOther recent applications of using polyhedron modeling with the concepts of Fractal Geometry involve studies on concrete rupture loads (Kotov, Volchuk, Zeziukov, & Pavlenko, 2023), fractal modeling for water treatment (Lorevice et al., 2023) and modeling of complex sets of viruses for the molecules encaps","cbCaiqOE9Nj5Y6Jt","https://ap.wps.com/l/cbCaiqOE9Nj5Y6Jt","pdf",750761,14,"English","# Introduction\n## 3D learning and immersive technologies\n## VR applications in education\n## Polyhedra and 3D fractal concepts\n## Related applications\n## Proposed VR/AR approach\n# Article objective","[{\"question\":\"Which geometric transformations are used during modeling and virtual room construction?\",\"answer\":\"The work uses homothety, rotation, and translation transformations applied to HTML page hierarchy structures in both phases.\"}]","Development of Virtual Reality Environments to Visualize the Fractals of the Dragon’s Curve Using Plato’s Polyhedra | PDF",35]