[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-588137-105":3,"detail-sidebar-cat-0-en-105":79,"doc-detail-588137-en":129},{"code":4,"msg":5,"data":6},0,"ok",{"site_id":7,"language":8,"slug":9,"title":10,"keywords":11,"description":12,"schema_data":13,"social_meta":72,"head_meta":74,"extra_data":76,"updated_unix":78},105,"en","plane-and-space-figurate-numbers-visualization-geogebra-help","PLANE AND SPACE FIGURATE NUMBERS: VISUALIZATION - GeoGebra Help","","The study and interest in figurate numbers trace back to ancient Greece, while their historical evolution and generalization of 2D, 3D, and m-D figurate numbers continue to motivate current research. The work connects number theory with mathematics education by emphasizing visualization for teaching. It proposes a discussion framework in which arithmetic, algebraic, and geometric properties become easier to understand using GeoGebra, supported by the mathematical teacher’s guided use.",{"@graph":14,"@context":71},[15,34,54],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/document/","Document",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/document/technology/","Technology",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/document/plane-and-space-figurate-numbers-visualization-geogebra-help/588137/",4,{"url":32,"name":10,"@type":35,"image":36,"author":41,"headline":10,"publisher":44,"fileFormat":47,"inLanguage":8,"description":12,"dateModified":48,"datePublished":48,"encodingFormat":47,"isAccessibleForFree":49,"interactionStatistic":50},"DigitalDocument",{"url":37,"@type":38,"width":39,"height":40},"https://docshare.wps.com/thumbnails/plane-and-space-figurate-numbers-visualization-geogebra-help/588137.png","ImageObject",300,407,{"name":42,"@type":43},"Aditya","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-10-15",true,{"@type":51,"interactionType":52,"userInteractionCount":4},"InteractionCounter",{"@type":53},"ViewAction",{"@type":55,"mainEntity":56},"FAQPage",[57,63,67],{"name":58,"@type":59,"acceptedAnswer":60},"Why are figurate numbers important for mathematics education?","Question",{"text":61,"@type":62},"The text highlights that, in Brazil, visualization becomes increasingly relevant when teaching mathematics, and figurate-number discussions can support that goal.","Answer",{"name":64,"@type":59,"acceptedAnswer":65},"How does GeoGebra support learning of figurate numbers?",{"text":66,"@type":62},"The proposal is to facilitate understanding of arithmetic, algebraic, and geometric properties through the use of GeoGebra software by the math teacher.",{"name":68,"@type":59,"acceptedAnswer":69},"What examples of 2D figurate numbers are constructed in the text?",{"text":70,"@type":62},"The document describes triangular numbers, square numbers, pentagonal numbers, and hexagonal numbers using point-addition constructions and regular polygon organization.","https://schema.org",{"og:url":32,"og:type":73,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":75,"canonical":32},"index,follow",{"doc_id":77,"site_id":7},588137,1792064602,{"code":4,"msg":80,"data":81},"success",[82,86,90,94,99,103,108,113,118,121,125],{"id":22,"doc_module":4,"doc_module_name":25,"category_name":83,"show_sort_weight":84,"slug":85},"Story & Novel",90,"story-novel",{"id":26,"doc_module":4,"doc_module_name":25,"category_name":87,"show_sort_weight":88,"slug":89},"Literature",80,"literature",{"id":33,"doc_module":4,"doc_module_name":25,"category_name":91,"show_sort_weight":92,"slug":93},"Exam",70,"exam",{"id":95,"doc_module":4,"doc_module_name":25,"category_name":96,"show_sort_weight":97,"slug":98},5,"Comic",60,"comic",{"id":100,"doc_module":4,"doc_module_name":25,"category_name":29,"show_sort_weight":101,"slug":102},6,50,"technology",{"id":104,"doc_module":4,"doc_module_name":25,"category_name":105,"show_sort_weight":106,"slug":107},7,"Healthcare",40,"healthcare",{"id":109,"doc_module":4,"doc_module_name":25,"category_name":110,"show_sort_weight":111,"slug":112},8,"Research & Report",30,"research-report",{"id":114,"doc_module":4,"doc_module_name":25,"category_name":115,"show_sort_weight":116,"slug":117},9,"Religion & Spirituality",20,"religion-spirituality",{"id":116,"doc_module":4,"doc_module_name":25,"category_name":119,"show_sort_weight":116,"slug":120},"World Cup","world-cup",{"id":122,"doc_module":4,"doc_module_name":25,"category_name":123,"show_sort_weight":122,"slug":124},10,"Lifestyle","lifestyle",{"id":126,"doc_module":4,"doc_module_name":25,"category_name":127,"show_sort_weight":95,"slug":128},19,"General","general",{"code":4,"msg":80,"data":130},{"doc_id":77,"user_id":131,"nickname":42,"user_avatar":132,"doc_module":4,"category_id":100,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":133,"file_id":134,"file_url":135,"file_type":136,"file_size":137,"view_count":4,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":138,"language":139,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":140,"faqs":141,"seo_title":142,"seo_description":12,"update_tm":78,"read_time":143},962085564549,"https://ap-avatar.wpscdn.com/davatar_085a072bc5b1113ac321206ff7593b45","PLANE AND SPACE FIGURATE NUMBERS: VISUALIZATION  \nWITH THE GEOGEBRA´ S HELP  \nFrancisco Regis Vieira Alves, Francisco Evamar Barros  \nAbstract: The study and interest of figurate numbers can be observed even in ancient Greece. On the other hand, it becomes very important the historical understanding about an evolutionary process and the particular generalization of such 2D, 3D and m-D figurate numbers and they remain the interest of current scientific investigations. On the other hand, in Brazil, when we talk about Mathematics Education, the component of visualization acquires more and more relevance for the teaching. In this way, in the present work, we present a proposal of discussion of the figure numbers, with idea that the understanding of arithmetic, algebraic and geometric properties can be facilitated with the use of GeoGebra software and it´s use by the mathematical teacher.  \nKey words: Figurate numbers 2D and 3D, Visualization, Mathematics Teaching, Mathematics Education.  \n1. Introduction  \nThe mathematical historian Eves (1969: 53) explains that \"the ancient Greeks distinguished between the study of conceptual relations between numbers and the practical art of counting.\" The author recalls that the Pythagoreans spoke about the numbers that allegedly possessed mystical properties, such as perfect numbers, deficient numbers, and abundant numbers; however, in this section, we will focus on the study of figurate which, according to Eves (1969, 54) originated with the first members of society Pythagorean. In essence, a figurate number presents the links between Arithmetic and Geometry, and other aspects are commented by Eves (1969, 1983), Dickson (1928) and Katz (1998) .  \nIn it´s turn, Popper (1972, p. 106) mentions without further explanation that Greek reasoning applied to forms geometrically, was extended to solids, despite the difficulties of identifying the 3D configurations which suggests the discussion of pyramidal numbers, or more specifically, pyramidal numbers triangles, square pyramidal numbers, pentagonal pyramidal numbers (Koshy, 2007) .  \nDeza & Deza (2012) comment on the interest of ancient Greek mathematicians about the figurate numbers. The theory of figurate numbers does not belong to the central domains of Mathematics, but the beauty of these numbers attracted the attention of many scientists over the years. In this sense, the authors comment afterward on an extensive list of mathematicians who have worked in this field.  \nThe list (not full) of famous mathematicians, who worked in this domain, consists of Pythagoras of Samos (ca. 582 BC–ca. 507 BC), Hypsicles of Alexandria (190 BC–120 BC), Plutarch of Chaeronea (ca. 46–ca. 122), Nicomachus of Gerasa (ca. 60–ca. 120), Theon of Smyrna (70– 135), Diophantus of Alexandria (ca. 210–ca. 290), Leonardo of Pisa, also known as Leonardo Fibonacci (ca. 1170 –ca. 1250), Michel Stifel (1487–1567), Gerolamo Cardano (1501–1576), Claude Gaspard Bachet de M´eziriac (1581–1638), René Descartes (1596–1650), Pierre de Fermat (1601–1665), John Pell (1611–1685), Blaise Pascal (1623–1662), Leonhard Euler (1707–1783), Joseph-Louis Lagrange (1736–1813), Adrien-Marie Legendre (1752–1833), Carl Friedrich Gauss (1777–1855), Augustin-Louis Cauchy (1789–1857), Carl Gustav Jacob Jacobi (1804–1851),Wac_law Franciszek Sierpinski (1882–1969), Barnes Wallis (1887–1979) . (Deza & Deza, 2012, p. Xvii) .  \nIn addition, several other formal theorems can be formulated from the figure numbers and they are closely related to special classes of other numbers and numerical sequences, such as Fibonacci-Lucas numbers, Mersenne and Fermat numbers (Wenchang, 2007) . Thus, in the field of Number Theory, we  \ncan find innumerable generalizations. On the other hand, we observe the relationships of figurate numbers with important theorems still in the present day, as expressed below Deza & Deza (2012) .  \nFigurate numbers were studied by the ancients, as far back as the Pythagoreans, but nowadays th","cbCaivlt24Nvzpdh","https://ap.wps.com/l/cbCaivlt24Nvzpdh","pdf",2138209,18,"English","# Introduction\n## Historical background and figurate numbers\n## Construction of 2D figurate numbers\n## Visualization and related theorems","[{\"question\":\"Why are figurate numbers important for mathematics education?\",\"answer\":\"The text highlights that, in Brazil, visualization becomes increasingly relevant when teaching mathematics, and figurate-number discussions can support that goal.\"},{\"question\":\"How does GeoGebra support learning of figurate numbers?\",\"answer\":\"The proposal is to facilitate understanding of arithmetic, algebraic, and geometric properties through the use of GeoGebra software by the math teacher.\"},{\"question\":\"What examples of 2D figurate numbers are constructed in the text?\",\"answer\":\"The document describes triangular numbers, square numbers, pentagonal numbers, and hexagonal numbers using point-addition constructions and regular polygon organization.\"}]","PLANE AND SPACE FIGURATE NUMBERS: VISUALIZATION - GeoGebra Help | PDF",45]