{"id":5,"date":"2019-02-08T20:52:32","date_gmt":"2019-02-08T20:52:32","guid":{"rendered":"https:\/\/martyknop.com\/blog\/?p=5"},"modified":"2025-10-05T05:22:28","modified_gmt":"2025-10-05T05:22:28","slug":"concentric-polygons","status":"publish","type":"post","link":"https:\/\/martyknop.com\/blog\/concentric-polygons\/","title":{"rendered":"CONCENTRIC POLYGONS"},"content":{"rendered":"\n<style>\nnav.catNavWrap a:nth-child(2) span{\n   background-color: yellow;\n}\n<\/style>\n<h3 class=\"blogtitletop\">TRIANGULAR NUMBERS<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"400\" height=\"267\" src=\"https:\/\/martyknop.com\/blog\/wp-content\/uploads\/2019\/02\/diagram_707.svg\" alt=\"\" class=\"wp-image-136\"\/><\/figure>\n<\/div>\n\n\n<p>  Triangular&nbsp;numbers are equal to the sum of the natural numbers. The next&nbsp;triangular number is generated by adding the next natural number to the base of the triangle.  <\/p>\n\n\n\n<h3 class=\"blogtitle\">CONCENTRIC POLYGONS<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/martyknop.com\/blog\/wp-content\/uploads\/2019\/02\/b_blog_002x4.svg\" alt=\"\" class=\"wp-image-21\"\/><\/figure>\n<\/div>\n\n\n<p>\n\n Inscribed within a set of concentric circles, this series of regular polygons begins with a single chord inscribed at the innermost circle then expands outwards always increasing the number of sides of the next polygon by 1.\n\n<\/p>\n\n\n\n<h3 class=\"blogtitle\">CORRELATION<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/martyknop.com\/blog\/wp-content\/uploads\/2019\/02\/b_blog_003x2.svg\" alt=\"\" class=\"wp-image-24\"\/><\/figure>\n<\/div>\n\n\n<p>  Each successive polygon is inscribed with its vertex facing right. Each node is numbered starting from its vertex and rotating in a counterclockwise direction. Triangular numbered nodes result as each successive ring increases by the amount of the next polygon.  <\/p>\n\n\n\n<h3 class=\"blogtitle\">THE FIRST 629 POINTS<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/martyknop.com\/blog\/wp-content\/uploads\/2019\/02\/b_blog_004x3.svg\" alt=\"\" class=\"wp-image-22\"\/><\/figure>\n<\/div>\n\n\n<p>  Above are the first 629 points inscribed within 34 concentric polygons. The vertex of each polygon line up to form a row of&nbsp;triangular&nbsp;numbered nodes.<\/p>\n\n\n<p><br \/>\n<\/p>\n\n\n\n\n<h3 class=\"blogtitletop\">PRONIC AND SQUARE NUMBERS<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/martyknop.com\/blog\/wp-content\/uploads\/2019\/02\/b_blog_009x3.svg\" alt=\"\" class=\"wp-image-25\"\/><\/figure>\n<\/div>\n\n\n<p>\n\nPronic numbers are equal to the progressive sum of even numbers whereas&nbsp;square numbers are equal to the progressive sum of odd numbers. \n\n<\/p>\n\n\n\n<h3 class=\"blogtitle\">EVEN-SIDED POLYGONS<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/martyknop.com\/blog\/wp-content\/uploads\/2019\/02\/b_blog_010.svg\" alt=\"\" class=\"wp-image-26\"\/><\/figure>\n<\/div>\n\n\n<p>\n\nIn this series each successive polygon has an even number of sides. The progression between each ring is 2 nodes and the arrangement is reflected across the x and y axis. \n\n<\/p>\n\n\n\n<h3 class=\"blogtitle\">CORRELATION<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/martyknop.com\/blog\/wp-content\/uploads\/2019\/02\/b_blog_011x2.svg\" alt=\"\" class=\"wp-image-27\"\/><\/figure>\n<\/div>\n\n\n<p>\n\n Again, each successive polygon is inscribed with its vertex facing right. Each ring is numbered counterclockwise bisected on the right side with a&nbsp;square&nbsp;number and ending on the left side with a&nbsp;pronic. \n\n<\/p>\n\n\n\n<h3 class=\"blogtitle\">THE FIRST 650 POINTS<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/martyknop.com\/blog\/wp-content\/uploads\/2019\/02\/b_blog_012x3.svg\" alt=\"\" class=\"wp-image-28\"\/><\/figure>\n<\/div>\n\n\n<p>  Above are the first 650 points inscribed within 25 concentric even-sided polygons. The horizontal axis is split between&nbsp;square&nbsp;numbers on the right and&nbsp;pronics&nbsp;to the left.  <\/p>\n\n\n<p><br \/>\n<\/p>\n\n\n\n\n<h3 class=\"blogtitletop\">TRIANGULAR SQUARE AND TRIANGULAR PRONIC NUMBERS<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"400\" height=\"269\" src=\"https:\/\/martyknop.com\/blog\/wp-content\/uploads\/2025\/04\/b_blog_017x3_02.svg\" alt=\"\" class=\"wp-image-161\" style=\"width:560px\"\/><\/figure>\n<\/div>\n\n\n<p>\n\n Numbers which are both triangular and square are called triangular squares, numbers which are both triangular and pronic are called triangular pronics. \n\n<\/p>\n\n\n\n<h3 class=\"blogtitle\">TRIANGULAR, SQUARE, AND PRONIC<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"400\" height=\"568\" src=\"https:\/\/martyknop.com\/blog\/wp-content\/uploads\/2019\/02\/b_blog_016x3_002.svg\" alt=\"\" class=\"wp-image-164\" style=\"width:560px\"\/><\/figure>\n<\/div>\n\n\n<p>  Inputting the next counting number into the formulae above outputs the next number in either the triangular, square, or pronic sequence. The respective m-valued or n-valued number correlates to the sequential order of the term in the series.  <\/p>\n\n\n\n<h3 class=\"blogtitle\">ALTERNATING SEQUENCES<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/martyknop.com\/blog\/wp-content\/uploads\/2019\/02\/b_blog_018x.svg\" alt=\"\" class=\"wp-image-33\"\/><\/figure>\n<\/div>\n\n\n<p>  Above is a diagram which maps out the first 4 triangular square and triangular pronic numbers as well as their respective m and n-values. These two series when put in ascending order, alternate back and forth between triangular square and triangular pronic values.  <\/p>\n\n\n\n<h3 class=\"blogtitle\">THE FIRST 49 TRIANGULAR NUMBERS<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/martyknop.com\/blog\/wp-content\/uploads\/2019\/02\/b_blog_014x3.svg\" alt=\"\" class=\"wp-image-29\"\/><\/figure>\n<\/div>\n\n\n<p>  Above are the first 49 triangular numbers inscribed within 35 concentric even-sided polygons. While pronic and square numbers line up in a row, triangular numbers form a galaxy-like spiral.  <\/p>\n\n\n<p><br \/>\n<\/p>\n","protected":false},"excerpt":{"rendered":"<p>TRIANGULAR NUMBERS Triangular&nbsp;numbers are equal to the sum of the natural numbers. The next&nbsp;triangular number is generated by adding the next natural number to the base of the triangle. CONCENTRIC POLYGONS Inscribed within a set of concentric circles, this series of regular polygons begins with a single chord inscribed at the innermost circle then expands [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-5","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/martyknop.com\/blog\/wp-json\/wp\/v2\/posts\/5","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/martyknop.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/martyknop.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/martyknop.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/martyknop.com\/blog\/wp-json\/wp\/v2\/comments?post=5"}],"version-history":[{"count":18,"href":"https:\/\/martyknop.com\/blog\/wp-json\/wp\/v2\/posts\/5\/revisions"}],"predecessor-version":[{"id":182,"href":"https:\/\/martyknop.com\/blog\/wp-json\/wp\/v2\/posts\/5\/revisions\/182"}],"wp:attachment":[{"href":"https:\/\/martyknop.com\/blog\/wp-json\/wp\/v2\/media?parent=5"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/martyknop.com\/blog\/wp-json\/wp\/v2\/categories?post=5"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/martyknop.com\/blog\/wp-json\/wp\/v2\/tags?post=5"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}