Figure 2
The Knuth Twin Dragon K and its intersection with 1;0;r for some r as in Theorem 3.1 (red) and with 1;0;1=5 (blue).
Source publication
The Knuth Twin Dragon is a compact subset of the plane with fractal boundary of Hausdorff dimension s = (\log \lambda)/(\log \sqrt{2}) , \lambda^{3} = \lambda^{2} + 2 . Although the intersection with a generic line has Hausdorff dimension s-1 , we prove that this does not occur for lines with rational parameters. We further describe the intersectio...
Context in source publication
Context 1
... go on with 1;0;1=4C1=16 and see that points in the intersection have imaginary part with an expansion in base 4 starting with two digits in ¹¹2; 0; 1; 3º and ending with digits in ¹¹1; 0; 1; 2º. For the limit 1;0;1=5 of lines of this form, we obtain the following intersection with K, see Figure 2. ...
Citations
... The Dragon Fractal, also called the Dragon curve, Heighway curve, or Jurassic Park Dragon, is a popular selfsimilar shape that appears in the book Jurassic Park by Michael Crichton. The properties of this fractal began to be investigated by NASA physicists John Heighway, Bruce Banks and William Harter and were described by Martin Gardner in the American scientific column Mathematical Games in 1967 (Großkopf, 2020;Kamiya, 2022). ...
... Figure 1 shows the first 10 iterations of this fractal construction from a straight line segment with measure a. Figure 1. Construction of the Dragon Fractal with 10 iterations Source: Großkopf, 2020. In each iteration, the measurement of segment a undergoes a reduction corresponding to the right-angled triangle formation, with sides equal to b and hypotenuse of measurement a. ...
This 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.
... Os ambientes virtuais programados em RV podem auxiliar nos estudos da Geometria Fractal, pois os alunos podem interagir e visualizar os sólidos e suas propriedades de maneira mais efetiva e significativa (Cangas et al., 2021) (Großkopf, 2020;Kamiya, 2022). ...
Resumo:
Neste artigo são apresentadas as adaptações para a criação de um conjunto de fractais da curva do dragão utilizando os poliedros de Platão. Os fractais modelados foram inseridos em ambientes programados com recursos da Realidade Virtual (RV), que permitem ao visitante a manipulação e visualização de cada iteração usada para construção destes fractais. As estruturas de hierarquias de páginas HTML foram usadas por meio das transformações geométricas de translação, homotetia e rotação nas duas fases deste trabalho: nas modelagens dos fractais e também nas construções das salas virtuais. Os recursos apresentados neste artigo podem ser usados em sala de aula para a visualização dos fractais de poliedros com óculos imersivos e até mesmo em Realidade Aumentada (RA) com o uso de smartphones ou tablets. O objetivo deste artigo é de mostrar o uso de tecnologias simples e gratuitas, que podem ser usadas para a criação de materiais didáticos com grande contribuição para a melhoria do ensino da Geometria Fractal, além de outras áreas que utilizam representações gráficas de objetos 3D. Palavras-chave: realidade virtual; fractais; poliedros de Platão; curva do dragão.
Abstract:
This article presents the adaptations for creating a set of Dragon Curve fractals using the Plato's 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 the geometric transformations of translation, homothety and rotation in both phases of this work: in the fractals modeling and also in the virtual rooms construction. The resources presented in this article can be used in the classroom to visualize polyhedra fractals with immersive glasses and even in 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.
