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We construct genomic predictors for heritable but extremely complex human quantitative traits (height, heel bone density, and educational attainment) using modern methods in high dimensional statistics (i.e., machine learning). The constructed predictors explain, respectively, ~40, 20, and 9% of total variance for the three traits, in data not used for training. For example, predicted heights correlate ~0.65 with actual height; actual heights of most individuals in validation samples are within a few centimeters of the prediction. The proportion of variance explained for height is comparable to the estimated common SNP heritability from genome-wide complex trait analysis (GCTA), and seems to be close to its asymptotic value (i.e., as sample size goes to infinity), suggesting that we have captured most of the heritability for SNPs. Thus, our results close the gap between prediction R-squared and common SNP heritability. The ~20k activated SNPs in our height predictor reveal the genetic architecture of human height, at least for common variants. Our primary dataset is the UK Biobank cohort, comprised of almost 500k individual genotypes with multiple phenotypes. We also use other datasets and SNPs found in earlier genome-wide association studies (GWAS) for out-of-sample validation of our results.