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=Robinson Triangles= | =Robinson Triangles= | ||
| + | Robinson triangles, also known as golden triangles due to their close relation with the golden ratio discussed previously, make up the spikes of regular pentagrams. | ||
| + | |||
| + | <center> | ||
| + | [[File:Golden star.png|frame|300x250px|Pentagram]] <br> | ||
| + | <small> [https://en.wikipedia.org/wiki/Golden_triangle_(mathematics) Wikipedia] </small> | ||
| + | </center> | ||
| + | |||
| + | A golden triangle is an isosceles triangle where a ratio between one of the identical sides <math> a </math> and the base <math> b </math> is the golden ratio <math> \phi </math>. | ||
| + | |||
| + | <center> <math> \frac {a}{b} = \phi </math> </center> | ||
| + | |||
| + | <center> | ||
| + | [[File:Golden triangle.png|frame|300x250px|Golden triangle]] <br> | ||
| + | <small> [https://en.wikipedia.org/wiki/Golden_triangle_(mathematics) Wikipedia] </small> | ||
| + | </center> | ||
| + | |||
| + | A similar triangle to the Robinson triangle is the golden gnomon: | ||
| + | |||
| + | <center> | ||
| + | [[File:Golden gnomom.jpg|frame|300x250px|Golden gnomon]] <br> | ||
| + | <small> [https://en.wikipedia.org/wiki/Golden_triangle_(mathematics) Wikipedia] </small> | ||
| + | </center> | ||
| + | |||
| + | The golden gnomon is another isosceles triangle where a ratio between one of the identical sides <math> a </math> and the base <math> b </math> is the reciprocal of the golden ratio <math> \frac{1}{\phi} </math>. | ||
| + | |||
| + | Regarding Penrose tiling, a golden triangle and two golden gnomons make up a regular pentagon, which ties in with the golden ratio local pentagon symmetry discussed earlier. | ||
| + | P2 Penrose tiling are made from kites and darts. A kite is made from two golden triangles, and a dart is made from two gnomons. | ||
| + | |||
| + | ==Further Readings:== | ||
| + | One final note – if you like proofs, you will enjoy this site: http://mrbertman.com/penroseTilings.html. It contains many definitions and theorems that deal with how to place a tile correctly (the site uses the term “legally”) and why those rules exist. You can even create your own Penrose tiling! | ||
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[[MA271Fall2020Walther_Topic27_Golden Ratio|Previous Section: Golden Ratio]] | [[MA271Fall2020Walther_Topic27_Golden Ratio|Previous Section: Golden Ratio]] | ||
| − | [[ | + | [[MA271Fall2020Walther_Topic27_Real World Examples|Next Section: Real World Examples]] |
[[Category:MA271Fall2020Walther]] | [[Category:MA271Fall2020Walther]] | ||
Latest revision as of 00:11, 7 December 2020
Robinson Triangles
Robinson triangles, also known as golden triangles due to their close relation with the golden ratio discussed previously, make up the spikes of regular pentagrams.
A golden triangle is an isosceles triangle where a ratio between one of the identical sides $ a $ and the base $ b $ is the golden ratio $ \phi $.
A similar triangle to the Robinson triangle is the golden gnomon:
The golden gnomon is another isosceles triangle where a ratio between one of the identical sides $ a $ and the base $ b $ is the reciprocal of the golden ratio $ \frac{1}{\phi} $.
Regarding Penrose tiling, a golden triangle and two golden gnomons make up a regular pentagon, which ties in with the golden ratio local pentagon symmetry discussed earlier. P2 Penrose tiling are made from kites and darts. A kite is made from two golden triangles, and a dart is made from two gnomons.
Further Readings:
One final note – if you like proofs, you will enjoy this site: http://mrbertman.com/penroseTilings.html. It contains many definitions and theorems that deal with how to place a tile correctly (the site uses the term “legally”) and why those rules exist. You can even create your own Penrose tiling!
