text: ## Polygons Related to the Golden Ratio, and Associated Figures in Geometry, Part 1: Triangles
  
  There are two isosceles triangles which are related to the golden ratio, [1 + sqrt(5)]/2, and I used to refer to them as the “golden acute isosceles triangle” and the “golden obtuse isosceles triangle,” before I found out these triangles already have other names –the ones shown above. The golden triangle, especially, shows up in some well-known polyhedra, such as both the great and small stellated dodecahedron. The triangles which form the “points” or “arms” of regular star pentagons (also known as pentagrams) are also golden triangles.
  
  These triangles have sides which are in the golden ratio. For the golden triangle, it is the leg:base ratio which is golden, as shown above. For the golden gnomon, this ratio is reversed:  the base:leg ratio is φ, or ~1.61803 — the irrational number known as the golden ratio.
  
  The angle ratios of each of these triangles are also unique. The golden triangle’s angles are in a 1:2:2 ratio, while the angles of the golden gnomon are in a ratio of 1:1:3.
  
  Another interesting fact about these two triangles is that each one can be subdivided into one of each type of triangle. The golden triangle can be split into a golden gnomon, and a smaller golden triangle, while the golden gnomon can be split into a golden triangle, and a smaller golden gnomon, as seen below.
  
  This process can be repeated indefinitely, in each case, creating ever-smaller triangles of each type.
  
  Polyhedra which use these triangles, as either faces or “facelets” (the visible parts of partially-hidden faces) are not uncommon, as previously mentioned. The three most well-known examples are three of the four Kepler-Poinsot solids. In the first two shown below, the small stellated dodecahedron and the great stellated dodecahedron, the actual faces are regular star pentagons which interpenetrate, but the facelets are golden triangles.
  
  The next example is also a Kepler-Poinsot solid: the great dodecahedron. Its actual faces are simply regular pentagons, not star pentagons, but, again, they interpenetrate, hiding much of each face from view. The visible parts, or “facelets,” are golden gnomons.
  
  For another example of a polyhedron made of golden gnomons, I made one myself — meaning that if anyone else has ever seen this polyhedron before, this fact is unknown to me, although I cannot rule it out. I have not given it a name. It has thirty-six faces, all of which are golden gnomons. There are twelve of the larger ones, shown in yellow, and twenty-four of the smaller ones, shown in red. This polyhedron has pyritohedral symmetry (the same type of symmetry seen in the seam-pattern of a typical volleyball), and its convex hull is the icosahedron.
  
  [Picture credits: to create the images in this post, I used both Geometer’s Sketchpad and MS-Paint for the two still, flat pictures found at the top. To make images of the four rotating polyhedra, I used a different program, Stella 4d: Polyhedron Navigator. Stella is available for purchase, with a free trial download available, at http://www.software3d.com/Stella.php.]
  
  I go by RobertLovesPi on-line, and am interested in many things. Welcome to my little slice of the Internet. The viewpoints and opinions expressed on this website are my own. They should not be confused with the views of my employer, nor any other organization, nor institution, of any kind.
  This entry was posted in Mathematics and tagged , , , , , , , , , , , . Bookmark the permalink.
  
  ### 3 Responses to Polygons Related to the Golden Ratio, and Associated Figures in Geometry, Part 1: Triangles
  
  1. howardat58 says:
  
  This is fascinating. An interesting spiral may be lurking in the decomposition somewhere.
  
  Liked by 1 person
  
  2. ivasallay says:
  
  I’ve never heard of the golden gnomon before. Thank you for enlightening me. As always, the solids you’ve created are awesome!
  
  Liked by 1 person
explanation: The document provides an in-depth exploration of the golden ratio in the context of isosceles triangles, specifically the golden triangle and the golden gnomon. It describes their unique properties, such as their side and angle ratios, and how they can be subdivided into smaller versions of each other. The text also discusses their appearance in various polyhedra, including the Kepler-Poinsot solids and a custom polyhedron created by the author. The document does not contain a specific question or answer.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: ## Polygons Related to the Golden Ratio, and Associated Figures in Geometry, Part 1: Triangles
  
  There are two isosceles triangles which are related to the golden ratio, [1 + sqrt(5)]/2, and I used to refer to them as the “golden acute isosceles triangle” and the “golden obtuse isosceles triangle,” before I found out these triangles already have other names –the ones shown above. The golden triangle, especially, shows up in some well-known polyhedra, such as both the great and small stellated dodecahedron. The triangles which form the “points” or “arms” of regular star pentagons (also known as pentagrams) are also golden triangles.
  
  These triangles have sides which are in the golden ratio. For the golden triangle, it is the leg:base ratio which is golden, as shown above. For the golden gnomon, this ratio is reversed:  the base:leg ratio is φ, or ~1.61803 — the irrational number known as the golden ratio.
  
  The angle ratios of each of these triangles are also unique. The golden triangle’s angles are in a 1:2:2 ratio, while the angles of the golden gnomon are in a ratio of 1:1:3.
  
  Another interesting fact about these two triangles is that each one can be subdivided into one of each type of triangle. The golden triangle can be split into a golden gnomon, and a smaller golden triangle, while the golden gnomon can be split into a golden triangle, and a smaller golden gnomon, as seen below.
  
  This process can be repeated indefinitely, in each case, creating ever-smaller triangles of each type.
  
  Polyhedra which use these triangles, as either faces or “facelets” (the visible parts of partially-hidden faces) are not uncommon, as previously mentioned. The three most well-known examples are three of the four Kepler-Poinsot solids. In the first two shown below, the small stellated dodecahedron and the great stellated dodecahedron, the actual faces are regular star pentagons which interpenetrate, but the facelets are golden triangles.
  
  The next example is also a Kepler-Poinsot solid: the great dodecahedron. Its actual faces are simply regular pentagons, not star pentagons, but, again, they interpenetrate, hiding much of each face from view. The visible parts, or “facelets,” are golden gnomons.
  
  For another example of a polyhedron made of golden gnomons, I made one myself — meaning that if anyone else has ever seen this polyhedron before, this fact is unknown to me, although I cannot rule it out. I have not given it a name. It has thirty-six faces, all of which are golden gnomons. There are twelve of the larger ones, shown in yellow, and twenty-four of the smaller ones, shown in red. This polyhedron has pyritohedral symmetry (the same type of symmetry seen in the seam-pattern of a typical volleyball), and its convex hull is the icosahedron.
  
  [Picture credits: to create the images in this post, I used both Geometer’s Sketchpad and MS-Paint for the two still, flat pictures found at the top. To make images of the four rotating polyhedra, I used a different program, Stella 4d: Polyhedron Navigator. Stella is available for purchase, with a free trial download available, at http://www.software3d.com/Stella.php.]
  
  I go by RobertLovesPi on-line, and am interested in many things. Welcome to my little slice of the Internet. The viewpoints and opinions expressed on this website are my own. They should not be confused with the views of my employer, nor any other organization, nor institution, of any kind.
  This entry was posted in Mathematics and tagged , , , , , , , , , , , . Bookmark the permalink.
  
  ### 3 Responses to Polygons Related to the Golden Ratio, and Associated Figures in Geometry, Part 1: Triangles
  
  1. howardat58 says:
  
  This is fascinating. An interesting spiral may be lurking in the decomposition somewhere.
  
  Liked by 1 person
  
  2. ivasallay says:
  
  I’ve never heard of the golden gnomon before. Thank you for enlightening me. As always, the solids you’ve created are awesome!
  
  Liked by 1 person
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document provides an in-depth exploration of the golden ratio in the context of isosceles triangles, specifically the golden triangle and the golden gnomon. It describes their unique properties, such as their side and angle ratios, and how they can be subdivided into smaller versions of each other. The text also discusses their appearance in various polyhedra, including the Kepler-Poinsot solids and a custom polyhedron created by the author. The document does not contain a specific question or answer.
  
  Label: no-question