Prior to the discovery of quasicrystals, crystals were modeled as discrete lattices, generated by a list of independent finite translations (Coxeter 1989). Because discreteness requires that the spacings between lattice points have a lower bound, the group of rotational symmetries of the lattice at any point must be a finite group. The strength of the theorem is that not all finite groups are compatible with a discrete lattice; in any dimension, we will have only a finite number of compatible groups.
The special cases of 2D (wallpaper groups) and 3D (space groups) are most heavily used in applications, and we can treat them together.
A shrinking argument also eliminates 5-fold symmetry. Consider a regular pentagon of lattice points. If it exists, then we can take every other edge displacement and (head-to-tail) assemble a 5-point star, with the last edge returning to the starting point. The vertices of such a star are again vertices of a regular pentagon with 5-fold symmetry, but about 60% smaller than the original.
The four translation vectors (three of which are given by r, and one which connects A' and B' given by r') form a parallelogram. Therefore, the length of r' is also given by:
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Matrix
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Matrix
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Matrix wallpaper 2 by
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the matrix wallpaper - Neo,
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The Matrix Has You Wallpapers
![](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_txXwLGaMKKFOdhe7OopyaPuNb3YkLsUPDWB5mgxstG_UU8y_ucS6J2hewzGlERitY7JiglH2ld4qUyRy2NoC-rnb4sOkQ6k_WP2WdB6t8axJ3Z897uY3x9FMX333sLHISVHlqqnWxvRrzvS9A=s0-d)
matrix wallpaper
![](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_tGbbIUlEMUlzMY1wOcO1I6FaWY1KHow_8vF9Ocg691uNxxM9ZWNNUAwQQs-Vo96wLU1dEY3oACPXrhJxk6CqJhIwcy9sg3fFJV8pOPRlj2obnPyen3t9Pddt0VCiq2_tnLVfUpH18IlodGNLKEB8CvhL6Q2ncu2sW63oVBe4OAVdz-JOpz5A=s0-d)
The Matrix Wallpaper
![](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_uastd6BWvdYtWCzwEnZlr5-InyudqBpweDvQIs7FMdgpMnQSOHDEypBHeykE02otFfd-bmoExtkQ8J6L1kARCdjGOBFMoAQBTIICTop2mTZ_NQ50wlBOCsyl8mb5K5qFtE6P9YX4uFeVaHwe0ylBPbzOZodds41NasNIr2gdGd9zqvn8bN2Qsl7z2pkAzQkQ=s0-d)
Wallpaper: Matrix
![](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_tN_lxrbvZ9ruBcJUmLUk_d1z2L4i1Lso58Ad8UTy75m18pbJISSJJ1lv_C7SsJpqKp2E2Hsc-8RVbdTBVw_YEQBToz39X7klvHrw-gH84N_IA8E-kP7vGoFREYjyS2TT6ZURGb=s0-d)
View \x26quot;The Matrix Wallpaper
![](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_t-oR23L1npJKT6ZGiFgc9NlerqZw8KfOQMCdRGHIsVlLFczO4W_goTVTg9gS_QUFGCXa2IKDh5Ezx-Tott4yAwWikP9B_41kZSmoCFucn9t4lHqorcfYwWKDejpzb8Anihrbtoa3P6IlVIdzC1AEL9xaSnJ8BhZhG9h64=s0-d)
Matrix Revolutions Wallpapers
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiZbn5mhly_VKzZ31oDyD4LbeywOB8kvQNLiljxmqfQBUbyiDAppdROk1pSJYu57VXduSvkeQFywC5qKJpgfcYRSjLSEU_IA5CAqC00TbUAQstFWJEr8skQn76EFFrO2dd9m2W2wc6sPcU/s1600/The+matrix+wallpapers.jpg)
The matrix wallpapers
![](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_tbrlmmbrusayHpTNrlsJiaRLLL46upw_PQlzCCGG76JSvpyqf7fRdtQWZw3BYvBJCvrfPavySv8c9k4DsN8f2ObkNWewWN6Mp0qKNQFc0HpKB7_XP0iYwzUvq0G4ZJZmEvco53=s0-d)
Matrix wallpapers
![](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_sAQPg5X75oQfFsOOt8p631aHNydp0hHyk7mKf9dbuVtN4KHjNQ2LapFAVsThHOUyds6v2nSYK7hoAYpU23Gc_5uraTPL6ztoHLHb4i-hCmPZcxnmz8IhP7dUKU9Sg1vidYIGkQ8Tb2_6aTasrC3PEzvOn-TA=s0-d)
The Simpsons Matrix Wallpaper
![](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vhRX2gVcBjs20yBtAdasPboyWcps7iMEetfY9pwwJoCBcBiaIW5x6TY7zE41XEty5NnlaXeVh8ByoftVPA-mFfBJeyvyCXnwiQh8PfoO4jzmdiYcr1At3pAuaMOs9Tm7ueEtPV0SW-zHhZV1WG1tF5xUSRy-mBf3CQ=s0-d)
Shawn Marion Matrix Wallpaper
![](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vRbqGwIxwNiPT7WiTRjmQm12vqUELT_gYNBwada-48-OyC2SYTlorhaUaZOVOfE0VinxlPKtIII1KFBz_bxlQUSAPQ9USd4CnAb5gyl3qS8-zCSbM=s0-d)
The following screenshot will
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Ubuntu and Matrix Wallpaper by
The special cases of 2D (wallpaper groups) and 3D (space groups) are most heavily used in applications, and we can treat them together.
A shrinking argument also eliminates 5-fold symmetry. Consider a regular pentagon of lattice points. If it exists, then we can take every other edge displacement and (head-to-tail) assemble a 5-point star, with the last edge returning to the starting point. The vertices of such a star are again vertices of a regular pentagon with 5-fold symmetry, but about 60% smaller than the original.
The four translation vectors (three of which are given by r, and one which connects A' and B' given by r') form a parallelogram. Therefore, the length of r' is also given by:
Matrix
Matrix
Matrix wallpaper 2 by
the matrix wallpaper - Neo,
The Matrix Has You Wallpapers
matrix wallpaper
The Matrix Wallpaper
Wallpaper: Matrix
View \x26quot;The Matrix Wallpaper
Matrix Revolutions Wallpapers
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiZbn5mhly_VKzZ31oDyD4LbeywOB8kvQNLiljxmqfQBUbyiDAppdROk1pSJYu57VXduSvkeQFywC5qKJpgfcYRSjLSEU_IA5CAqC00TbUAQstFWJEr8skQn76EFFrO2dd9m2W2wc6sPcU/s1600/The+matrix+wallpapers.jpg)
The matrix wallpapers
Matrix wallpapers
The Simpsons Matrix Wallpaper
Shawn Marion Matrix Wallpaper
The following screenshot will
Ubuntu and Matrix Wallpaper by
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