This page describes all points that appear on a unit ball for dimensions 1, 2, 3 and 4 where the decimal representation has a given fixed maximum number of non-zero digits for
We do not include points previously found in higher dimensions. You may permute and change signs as you wish.
This one is trivial: there is only one.
[1]
In addition to points on the axes (e.g., $(\pm 1, 0)$ and $(0,\pm 1)$), we also have the following. There is only one number per number of digits after the decimal point, so we present up to 13 digits after the decimal point.
The point $(0.70005137408, 0.71409248256)$ is tantalizingly close to $\left( \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}} \right)$.
[0.6, 0.8] [0.28, 0.96] [0.352, 0.936] [0.5376, 0.8432] [0.07584, 0.99712] [0.658944, 0.752192] [0.2063872, 0.9784704] [0.42197248, 0.90660864] [0.472103424, 0.881543168] [0.1512431616, 0.9884965888] [0.70005137408, 0.71409248256] [0.131585609728, 0.991304810496] [0.4895143985152, 0.8719952142336]
In addition to points on the axes (e.g., $(0,\pm 1,0)$) and points around the three principal great circles (e.g., $(\pm 0.6, 0, \pm 0.8)$ and $(\pm 0.96,\pm 0.28, 0 )$), we also have the following with two to five digits (there are none with just one) after the decimal point.
The point $(0.54144, 0.55808, 0.6288)$ is close to $\left( \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}} \right)$.
[0.36, 0.48, 0.8 ] [0.48, 0.6, 0.64] [0.024, 0.64, 0.768] [0.096, 0.36, 0.928] [0.096, 0.48, 0.872] [0.152, 0.48, 0.864] [0.168, 0.224, 0.96 ] [0.168, 0.576, 0.8 ] [0.192, 0.48, 0.856] [0.192, 0.64, 0.744] [0.224, 0.6, 0.768] [0.28, 0.576, 0.768] [0.352, 0.36, 0.864] [0.424, 0.48, 0.768] [0.0192, 0.168, 0.9856] [0.0192, 0.224, 0.9744] [0.024, 0.0512, 0.9984] [0.024, 0.2304, 0.9728] [0.0512, 0.6384, 0.768 ] [0.0656, 0.192, 0.9792] [0.0656, 0.4992, 0.864 ] [0.0768, 0.424, 0.9024] [0.0768, 0.4976, 0.864 ] [0.0784, 0.2688, 0.96 ] [0.096, 0.1392, 0.9856] [0.096, 0.2688, 0.9584] [0.096, 0.4096, 0.9072] [0.096, 0.5264, 0.8448] [0.1216, 0.48, 0.8688] [0.1216, 0.6288, 0.768 ] [0.1296, 0.192, 0.9728] [0.1296, 0.6272, 0.768 ] [0.1344, 0.152, 0.9792] [0.1344, 0.4992, 0.856 ] [0.1392, 0.3456, 0.928 ] [0.152, 0.4032, 0.9024] [0.1664, 0.4752, 0.864 ] [0.168, 0.2944, 0.9408] [0.168, 0.3968, 0.9024] [0.168, 0.6336, 0.7552] [0.1808, 0.6144, 0.768 ] [0.192, 0.2112, 0.9584] [0.192, 0.3968, 0.8976] [0.2112, 0.2816, 0.936 ] [0.2112, 0.4416, 0.872 ] [0.2112, 0.5616, 0.8 ] [0.224, 0.2544, 0.9408] [0.224, 0.4416, 0.8688] [0.224, 0.6672, 0.7104] [0.2304, 0.2928, 0.928 ] [0.2304, 0.352, 0.9072] [0.2304, 0.6272, 0.744 ] [0.2368, 0.36, 0.9024] [0.2368, 0.576, 0.7824] [0.2688, 0.28, 0.9216] [0.2688, 0.4416, 0.856 ] [0.2816, 0.6, 0.7488] [0.2928, 0.4096, 0.864 ] [0.2928, 0.5696, 0.768 ] [0.2928, 0.64, 0.7104] [0.3264, 0.3648, 0.872 ] [0.3264, 0.424, 0.8448] [0.3456, 0.5392, 0.768 ] [0.3456, 0.576, 0.7408] [0.352, 0.4752, 0.8064] [0.352, 0.5616, 0.7488] [0.36, 0.6336, 0.6848] [0.3648, 0.5264, 0.768 ] [0.3968, 0.48, 0.7824] [0.3968, 0.5376, 0.744 ] [0.4032, 0.4976, 0.768 ] [0.4416, 0.6288, 0.64 ] [0.4464, 0.48, 0.7552] [0.4464, 0.576, 0.6848] [0.48, 0.5696, 0.6672] [0.5392, 0.576, 0.6144] [0.00352, 0.59136, 0.8064 ] [0.00512, 0.2928, 0.95616] [0.00512, 0.66816, 0.744 ] [0.00672, 0.16896, 0.9856 ] [0.00672, 0.53504, 0.8448 ] [0.00864, 0.2368, 0.97152] [0.00864, 0.37248, 0.928 ] [0.01152, 0.05536, 0.9984 ] [0.01152, 0.23136, 0.9728 ] [0.01152, 0.50336, 0.864 ] [0.01152, 0.6848, 0.72864] [0.01632, 0.20224, 0.9792 ] [0.01632, 0.3456, 0.93824] [0.01632, 0.3968, 0.91776] [0.01632, 0.5696, 0.82176] [0.0192, 0.40544, 0.91392] [0.0192, 0.45696, 0.88928] [0.0192, 0.64512, 0.76384] [0.0192, 0.68768, 0.72576] [0.024, 0.39936, 0.91648] [0.024, 0.55808, 0.82944] [0.02656, 0.04992, 0.9984 ] [0.02656, 0.33792, 0.9408 ] [0.02656, 0.43008, 0.9024 ] [0.02688, 0.28416, 0.9584 ] [0.02688, 0.41984, 0.9072 ] [0.02944, 0.3456, 0.93792] [0.02944, 0.62208, 0.7824 ] [0.03072, 0.1808, 0.98304] [0.03072, 0.2112, 0.97696] [0.03072, 0.2816, 0.95904] [0.03072, 0.6288, 0.77696] [0.03968, 0.3648, 0.93024] [0.04096, 0.1296, 0.99072] [0.04096, 0.19872, 0.9792 ] [0.04224, 0.1216, 0.99168] [0.04224, 0.33632, 0.9408 ] [0.04224, 0.48768, 0.872 ] [0.04224, 0.6672, 0.74368] [0.04704, 0.38528, 0.9216 ] [0.04704, 0.63872, 0.768 ] [0.04992, 0.0512, 0.99744] [0.04992, 0.4464, 0.89344] [0.04992, 0.51456, 0.856 ] [0.0512, 0.23136, 0.97152] [0.0512, 0.57984, 0.81312] [0.0512, 0.61824, 0.78432] [0.05376, 0.16032, 0.9856 ] [0.05376, 0.168, 0.98432] [0.05376, 0.28032, 0.9584 ] [0.05376, 0.576, 0.81568] [0.05536, 0.2304, 0.97152] [0.05536, 0.48, 0.87552] [0.05536, 0.66048, 0.7488 ] [0.06272, 0.19296, 0.9792 ] [0.06272, 0.38304, 0.9216 ] [0.06272, 0.4032, 0.91296] [0.06272, 0.576, 0.81504] [0.0656, 0.11904, 0.99072] [0.0656, 0.39168, 0.91776] [0.0656, 0.43392, 0.89856] [0.0656, 0.66816, 0.74112] [0.06912, 0.33184, 0.9408 ] [0.06912, 0.4464, 0.89216] [0.06912, 0.65184, 0.7552 ] [0.06912, 0.66816, 0.7408 ] [0.07584, 0.21888, 0.9728 ] [0.0768, 0.12032, 0.98976] [0.0768, 0.20224, 0.97632] [0.0768, 0.39264, 0.91648] [0.0768, 0.46752, 0.88064] [0.0784, 0.36096, 0.92928] [0.0784, 0.60672, 0.79104] [0.08448, 0.27264, 0.9584 ] [0.08448, 0.43264, 0.8976 ] [0.08544, 0.14592, 0.9856 ] [0.08544, 0.3968, 0.91392] [0.08576, 0.096, 0.99168] [0.08576, 0.2928, 0.95232] [0.08576, 0.48768, 0.8688 ] [0.08576, 0.6144, 0.78432] [0.09216, 0.5392, 0.83712] [0.09216, 0.6848, 0.72288] [0.096, 0.21664, 0.97152] [0.096, 0.60544, 0.79008] [0.096, 0.70272, 0.70496] [0.09856, 0.26208, 0.96 ] [0.09856, 0.33792, 0.936 ] [0.10368, 0.352, 0.93024] [0.10368, 0.41824, 0.9024 ] [0.10752, 0.224, 0.96864] [0.10752, 0.63136, 0.768 ] [0.11424, 0.1344, 0.98432] [0.11424, 0.16768, 0.9792 ] [0.11616, 0.12288, 0.9856 ] [0.11616, 0.64512, 0.7552 ] [0.11904, 0.19072, 0.9744 ] [0.11904, 0.19872, 0.9728 ] [0.11904, 0.4416, 0.88928] [0.12032, 0.41376, 0.9024 ] [0.1216, 0.13728, 0.98304] [0.1216, 0.23712, 0.96384] [0.1216, 0.40704, 0.90528] [0.12288, 0.6336, 0.76384] [0.12896, 0.5376, 0.83328] [0.1296, 0.23808, 0.96256] [0.1296, 0.43008, 0.89344] [0.1296, 0.66304, 0.73728] [0.1344, 0.38528, 0.91296] [0.1344, 0.46592, 0.87456] [0.1344, 0.69216, 0.70912] [0.13472, 0.31104, 0.9408 ] [0.13472, 0.4416, 0.88704] [0.13472, 0.64896, 0.7488 ] [0.13728, 0.6144, 0.77696] [0.13824, 0.1664, 0.97632] [0.13824, 0.4976, 0.85632] [0.1392, 0.28032, 0.94976] [0.1392, 0.51456, 0.84608] [0.1392, 0.53504, 0.83328] [0.1392, 0.66816, 0.73088] [0.14336, 0.39552, 0.9072 ] [0.14336, 0.51552, 0.8448 ] [0.14592, 0.40544, 0.9024 ] [0.14592, 0.5392, 0.82944] [0.14592, 0.60544, 0.7824 ] [0.14976, 0.16768, 0.9744 ] [0.14976, 0.68768, 0.7104 ] [0.152, 0.21888, 0.96384] [0.152, 0.69504, 0.70272] [0.15616, 0.60288, 0.7824 ] [0.15616, 0.62112, 0.768 ] [0.15744, 0.16992, 0.9728 ] [0.15744, 0.23808, 0.9584 ] [0.16032, 0.2688, 0.94976] [0.1664, 0.40608, 0.89856] [0.16768, 0.3264, 0.93024] [0.16768, 0.36, 0.91776] [0.16768, 0.5616, 0.81024] [0.16768, 0.6672, 0.72576] [0.168, 0.32896, 0.92928] [0.168, 0.48384, 0.85888] [0.168, 0.60672, 0.77696] [0.16896, 0.38528, 0.9072 ] [0.16896, 0.5264, 0.83328] [0.16896, 0.56672, 0.8064 ] [0.16992, 0.2112, 0.96256] [0.1808, 0.24576, 0.95232] [0.18432, 0.2928, 0.93824] [0.18432, 0.5392, 0.82176] [0.19072, 0.2928, 0.93696] [0.19072, 0.45696, 0.8688 ] [0.192, 0.22112, 0.95616] [0.192, 0.40608, 0.89344] [0.192, 0.53504, 0.82272] [0.192, 0.68736, 0.70048] [0.19296, 0.31872, 0.928 ] [0.19296, 0.38528, 0.9024 ] [0.19296, 0.6848, 0.70272] [0.20224, 0.39168, 0.8976 ] [0.20224, 0.4464, 0.87168] [0.20224, 0.4752, 0.85632] [0.20224, 0.60768, 0.768 ] [0.2112, 0.30304, 0.92928] [0.2112, 0.33632, 0.91776] [0.2112, 0.42144, 0.88192] [0.2112, 0.43264, 0.87648] [0.2112, 0.57984, 0.78688] [0.2112, 0.65152, 0.72864] [0.21664, 0.37248, 0.9024 ] [0.21888, 0.2304, 0.94816] [0.21888, 0.44416, 0.8688 ] [0.22112, 0.66816, 0.7104 ] [0.224, 0.36096, 0.90528] [0.224, 0.43008, 0.87456] [0.224, 0.56928, 0.79104] [0.22656, 0.43392, 0.872 ] [0.22656, 0.5696, 0.79008] [0.2304, 0.26272, 0.93696] [0.2304, 0.32256, 0.91808] [0.2304, 0.51456, 0.82592] [0.2304, 0.56448, 0.79264] [0.2304, 0.56672, 0.79104] [0.2304, 0.60288, 0.76384] [0.23136, 0.3264, 0.91648] [0.23136, 0.36352, 0.9024 ] [0.2368, 0.25344, 0.93792] [0.2368, 0.28032, 0.93024] [0.2368, 0.50592, 0.82944] [0.23712, 0.44416, 0.864 ] [0.23808, 0.61856, 0.7488 ] [0.23808, 0.64, 0.73056] [0.23904, 0.2816, 0.92928] [0.23904, 0.47872, 0.8448 ] [0.24576, 0.424, 0.87168] [0.24576, 0.5696, 0.78432] [0.24576, 0.60768, 0.7552 ] [0.24704, 0.3456, 0.90528] [0.24704, 0.43872, 0.864 ] [0.24704, 0.6288, 0.73728] [0.24704, 0.6672, 0.70272] [0.25344, 0.62208, 0.7408 ] [0.2544, 0.38528, 0.88704] [0.2544, 0.61824, 0.74368] [0.26208, 0.352, 0.89856] [0.26208, 0.3648, 0.89344] [0.26208, 0.6272, 0.73344] [0.2688, 0.32896, 0.90528] [0.2688, 0.41984, 0.86688] [0.2688, 0.49824, 0.82432] [0.2688, 0.51328, 0.81504] [0.2688, 0.56928, 0.77696] [0.2688, 0.63872, 0.72096] [0.2688, 0.65184, 0.70912] [0.27008, 0.32256, 0.9072 ] [0.27264, 0.2928, 0.91648] [0.27264, 0.3968, 0.87648] [0.27264, 0.57952, 0.768 ] [0.28, 0.33792, 0.89856] [0.28032, 0.41824, 0.864 ] [0.2816, 0.39264, 0.87552] [0.2816, 0.62208, 0.73056] [0.28416, 0.4096, 0.86688] [0.29088, 0.3456, 0.89216] [0.29088, 0.66816, 0.6848 ] [0.2928, 0.37248, 0.88064] [0.2928, 0.44544, 0.84608] [0.2928, 0.60416, 0.74112] [0.29312, 0.31584, 0.9024 ] [0.2944, 0.43008, 0.85344] [0.2944, 0.65184, 0.69888] [0.29856, 0.3648, 0.88192] [0.29856, 0.4416, 0.84608] [0.31104, 0.38528, 0.8688 ] [0.31104, 0.5616, 0.76672] [0.31104, 0.58528, 0.7488 ] [0.31584, 0.4032, 0.85888] [0.31616, 0.4464, 0.83712] [0.31616, 0.6144, 0.72288] [0.31872, 0.40704, 0.856 ] [0.31872, 0.53504, 0.7824 ] [0.31872, 0.6336, 0.70496] [0.32256, 0.43008, 0.8432 ] [0.32256, 0.50592, 0.8] [0.3264, 0.42144, 0.84608] [0.3264, 0.47872, 0.81504] [0.33184, 0.6336, 0.69888] [0.33632, 0.48, 0.81024] [0.33632, 0.61824, 0.7104 ] [0.33792, 0.3968, 0.85344] [0.34208, 0.4416, 0.82944] [0.34208, 0.54144, 0.768 ] [0.3456, 0.44544, 0.82592] [0.3456, 0.65888, 0.66816] [0.352, 0.58752, 0.72864] [0.35424, 0.4416, 0.82432] [0.35712, 0.5376, 0.76384] [0.35712, 0.6272, 0.69216] [0.36, 0.57984, 0.73088] [0.36352, 0.39264, 0.8448 ] [0.36352, 0.56064, 0.744 ] [0.36352, 0.6288, 0.68736] [0.3648, 0.50176, 0.78432] [0.37248, 0.4976, 0.78336] [0.38304, 0.51328, 0.768 ] [0.38496, 0.3968, 0.83328] [0.38528, 0.4752, 0.79104] [0.38528, 0.576, 0.72096] [0.38784, 0.48, 0.78688] [0.38784, 0.5616, 0.73088] [0.39264, 0.64, 0.66048] [0.39552, 0.5264, 0.75264] [0.3968, 0.40704, 0.82272] [0.3968, 0.60288, 0.69216] [0.3968, 0.62112, 0.67584] [0.39936, 0.57952, 0.7104 ] [0.4032, 0.41984, 0.81312] [0.4032, 0.63072, 0.66304] [0.40608, 0.4992, 0.76544] [0.40608, 0.51456, 0.7552 ] [0.40704, 0.5392, 0.73728] [0.4096, 0.46752, 0.78336] [0.4096, 0.51552, 0.75264] [0.4096, 0.62112, 0.66816] [0.41984, 0.6384, 0.64512] [0.424, 0.60288, 0.67584] [0.43008, 0.47456, 0.768 ] [0.43008, 0.5264, 0.73344] [0.43008, 0.6, 0.67456] [0.43392, 0.4752, 0.76544] [0.43392, 0.60544, 0.6672 ] [0.43872, 0.5696, 0.69504] [0.4416, 0.52352, 0.72864] [0.4416, 0.56064, 0.70048] [0.4416, 0.57088, 0.69216] [0.45696, 0.4976, 0.73728] [0.45696, 0.6272, 0.63072] [0.46592, 0.4992, 0.73056] [0.4752, 0.59136, 0.65152] [0.48, 0.61856, 0.62208] [0.48384, 0.576, 0.65888] [0.4976, 0.57984, 0.64512] [0.4992, 0.57088, 0.65184] [0.4992, 0.60416, 0.62112] [0.50176, 0.58368, 0.6384 ] [0.50336, 0.58752, 0.6336 ] [0.50592, 0.5376, 0.67456] [0.50592, 0.60544, 0.6144 ] [0.51328, 0.5616, 0.64896] [0.5264, 0.58368, 0.61824] [0.53664, 0.56448, 0.6272 ] [0.54144, 0.55808, 0.6288 ]
In addition to points on the axes (e.g., $(0,0, \pm 1,0)$), points around the four principal great circles (e.g., $(\pm 0.6, 0, \pm 0.8)$ and $(\pm 0.96,\pm 0.28, 0 )$), points around the four principal great spheres (e.g., $(\pm 0.36, 0, \pm 0.48, \pm 0.8)$ and $(\pm 0.48, \pm 0.6, \pm 0.64, 0)$ ), we also have we also have the following with one, two or three digits after the decimal point.
[0.1, 0.1, 0.7, 0.7] [0.1, 0.3, 0.3, 0.9] [0.1, 0.5, 0.5, 0.7] [0.2, 0.4, 0.4, 0.8] [0.5, 0.5, 0.5, 0.5] [0.02, 0.02, 0.34, 0.94] [0.02, 0.1, 0.5, 0.86] [0.02, 0.14, 0.14, 0.98] [0.02, 0.14, 0.7, 0.7 ] [0.02, 0.22, 0.26, 0.94] [0.02, 0.22, 0.46, 0.86] [0.02, 0.26, 0.62, 0.74] [0.02, 0.34, 0.38, 0.86] [0.02, 0.34, 0.46, 0.82] [0.02, 0.34, 0.58, 0.74] [0.06, 0.06, 0.18, 0.98] [0.06, 0.06, 0.62, 0.78] [0.06, 0.1, 0.42, 0.9 ] [0.06, 0.18, 0.54, 0.82] [0.06, 0.42, 0.46, 0.78] [0.06, 0.42, 0.62, 0.66] [0.08, 0.08, 0.64, 0.76] [0.08, 0.12, 0.24, 0.96] [0.08, 0.16, 0.44, 0.88] [0.08, 0.2, 0.56, 0.8 ] [0.08, 0.24, 0.48, 0.84] [0.08, 0.32, 0.56, 0.76] [0.08, 0.4, 0.44, 0.8 ] [0.08, 0.52, 0.56, 0.64] [0.1, 0.1, 0.14, 0.98] [0.1, 0.26, 0.5, 0.82] [0.1, 0.3, 0.54, 0.78] [0.1, 0.34, 0.62, 0.7 ] [0.12, 0.24, 0.64, 0.72] [0.14, 0.22, 0.22, 0.94] [0.14, 0.22, 0.62, 0.74] [0.14, 0.46, 0.62, 0.62] [0.16, 0.16, 0.32, 0.92] [0.16, 0.2, 0.4, 0.88] [0.16, 0.32, 0.64, 0.68] [0.18, 0.26, 0.3, 0.9 ] [0.18, 0.26, 0.54, 0.78] [0.18, 0.54, 0.54, 0.62] [0.22, 0.26, 0.38, 0.86] [0.22, 0.26, 0.46, 0.82] [0.22, 0.26, 0.58, 0.74] [0.22, 0.46, 0.5, 0.7 ] [0.24, 0.44, 0.48, 0.72] [0.26, 0.34, 0.38, 0.82] [0.26, 0.46, 0.58, 0.62] [0.28, 0.32, 0.64, 0.64] [0.3, 0.3, 0.46, 0.78] [0.3, 0.3, 0.62, 0.66] [0.32, 0.4, 0.4, 0.76] [0.32, 0.52, 0.56, 0.56] [0.34, 0.34, 0.62, 0.62] [0.34, 0.38, 0.5, 0.7 ] [0.34, 0.46, 0.58, 0.58] [0.34, 0.5, 0.5, 0.62] [0.36, 0.48, 0.48, 0.64] [0.4, 0.4, 0.52, 0.64] [0.42, 0.42, 0.46, 0.66] [0.004, 0.004, 0.452, 0.892] [0.004, 0.02, 0.572, 0.82 ] [0.004, 0.028, 0.292, 0.956] [0.004, 0.028, 0.34, 0.94 ] [0.004, 0.028, 0.548, 0.836] [0.004, 0.044, 0.148, 0.988] [0.004, 0.068, 0.356, 0.932] [0.004, 0.068, 0.532, 0.844] [0.004, 0.076, 0.412, 0.908] [0.004, 0.092, 0.124, 0.988] [0.004, 0.092, 0.692, 0.716] [0.004, 0.1, 0.172, 0.98 ] [0.004, 0.116, 0.572, 0.812] [0.004, 0.124, 0.652, 0.748] [0.004, 0.14, 0.62, 0.772] [0.004, 0.148, 0.628, 0.764] [0.004, 0.172, 0.508, 0.844] [0.004, 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