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We study the spectral diversity of Type Ia supernovae (SNe Ia) at maximumlight using high signal-to-noise spectrophotometry of 173 SNe Ia from theNearby Supernova Factory. We decompose the diversity of these spectra intodifferent extrinsic and intrinsic components, and we construct a nonlinearparameterization of the intrinsic diversity of SNe Ia that preserves pairingsof "twin" SNe Ia. We call this parameterization the "Twins Embedding". Ourmethodology naturally handles highly nonlinear variability in spectra, such aschanges in the photosphere expansion velocity, and uses the full spectrumrather than being limited to specific spectral line strengths, ratios orvelocities. We find that the time evolution of SNe Ia near maximum light isremarkably similar, with 84.6% of the variance in common to all SNe Ia. Aftercorrecting for brightness and color, the intrinsic variability of SNe Ia ismostly restricted to specific spectral lines, and we find intrinsic dispersionsas low as ~0.02 mag between 6600 and 7200 A. With a nonlinear three-dimensionalmodel plus one dimension for color, we can explain 89.2% of the intrinsicdiversity in our sample of SNe Ia, which includes several different kinds of"peculiar" SNe Ia. A linear model requires seven dimensions to explain acomparable fraction of the intrinsic diversity. We show how a wide range ofpreviously-established indicators of diversity in SNe Ia can be recovered fromthe Twins Embedding. In a companion article, we discuss how these results an beapplied to standardization of SNe Ia for cosmology.

The Twins Embedding of Type Ia Supernovae I: The Diversity of Spectra at Maximum Light