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Chirality means that a molecule cannot be superposed on its mirror image. A carbon attached to four different groups is a common source of chirality; the definition concerns the whole molecule, rather than requiring this particular structural feature.

The test is spatial: apply a translation and a rotation to the entire molecule until corresponding chemical atoms coincide, keeping their identities and chemical bonds. A mirror operation is a reflection; physically turning a model is a rotation. They are different operations. If no such superposition exists, the two forms are enantiomers; if one succeeds, the molecule is achiral. A different-looking flat drawing alone establishes neither conclusion.

For a simple molecule with exactly one tetrahedral stereocenter and four constitutionally different substituents, the mirror forms cannot superpose. Compare entire attached groups, including paths around a ring, rather than just the first chemical atom. Several tetrahedral stereocenters require a whole-molecule check: a meso compound can be achiral. Conversely, axial chirality needs no tetrahedral stereocenter.^[ref]

Why care about this distinction? Two enantiomers can fit differently into an already chiral molecule, just as two opposite hands fit differently into one glove. Thus a protein's spatial arrangement can distinguish a mirror pair even when an achiral measurement cannot. Polarimetry provides a different kind of test: it measures a sample's net optical rotation, which depends on the populations of its enantiomers.

The separability of the printed Lagrangian is important. Every $x_i$ is an independent oscillator of the same frequency, irrespective of mass. Define $X_i=\sqrt{m_i}{x}_i$, $P_i=p_{x_i}/\sqrt{m_i}$. Then the complete constant matrix $Q_{ij}=P_i{P}_j+{\omega }^2{X}_i{X}_j$, and angular quantities $J_{ij}=X_i{P}_j-X_j{P}_i$, include each oscillator energy and the rotational symmetries of this mass-weighted oscillator space. Alternatively its explicitly time-dependent constants are $X_i\cos \omega t-P_i\sin \omega t/\omega$ and $X_i\sin \omega t+P_i\cos \omega t/\omega$.

Translations in all $y_i$ give conserved $P_y=\sum p_{y_i}$. Translation of $z_1,z_2$ together gives $P_{z12}=p_{z_1}+p_{z_2}$; $z_3$ is free and gives $p_{z_3}$. Their center-of-mass positions minus the associated constant velocities times $t$ are boost constants. The decoupled relative $z_1-z_2$ motion conserves $E_{z,\mathrm{rel}}=(p_{z_1}/m_1-p_{z_2}/m_2{)}^2{\mu }_{12}/2+V(z_1-z_2)$, where ${\mu }_{12}=m_1{m}_2/(m_1+m_2)$. In mass-weighted versions, the three free center coordinates also have conserved mutual angular momenta. These families are not all functionally independent.

The $y$ relative subsystem alone is driven through $\cos [\lambda (m_1-m_2)t]$. For generic nonzero $\lambda (m_1-m_2)$ its energy, and hence the total energy, is not conserved. Omitting the mass-weighted oscillator symmetries simply because the physical masses differ would miss genuine symmetries of this particular model.

DNA is a directional polymer with a sugar-phosphate backbone linked by phosphodiester bonds. Its deoxyribose sugars carry adenine, thymine, guanine or cytosine. In the usual cellular duplex, antiparallel strands form a double helix with complementary A–T and G–C pairing, stabilized by base stacking as well as hydrogen bonding. The sequence can vary without changing the basic backbone, allowing a large information capacity. Complementarity provides a template for copying and repair; the two strands are not merely two redundant lists stored without a copying mechanism. Major and minor grooves permit proteins to recognize sequence-dependent chemical patterns. Eukaryotic chromosomes package DNA around histones, while bacterial chromosomes are usually supercoiled and associated with other proteins; not all cellular DNA is nuclear or linear.

RNA uses ribose with a reactive 2-prime hydroxyl and usually uracil in place of thymine. It is often a single chain, but intramolecular complementary segments form stems, loops and elaborate three-dimensional folds. Messenger RNA conveys coding information to ribosomes; transfer RNA combines an anticodon with an amino-acid attachment site; ribosomal RNA has structural and catalytic roles in translation. Regulatory and catalytic RNAs show why it is inadequate to describe RNA solely as a transient DNA copy. RNA's local pairing and folding let sequence encode both information and molecular machinery. DNA's chemical stability and duplex repair make it especially suitable for long-term inheritance.

Replication is highly faithful but imperfect. Heritable sequence changes, recombination and gene duplication create variation, and selection, Genetic drift and population history determine its spread. Thus the same chemistry explains both continuity of lineages and their divergence; mutation is not directed toward what an organism needs. Conserved genes, homologous sequences and the widely shared genetic code support common ancestry. RNA catalysis makes an early RNA-based stage of evolution a plausible hypothesis, not a direct historical observation or a deduction that all modern RNA preceded DNA in a demonstrated sequence. Structure constrains function and evolutionary possibilities without uniquely determining a species' history.^[ref]

Take a central chemical atom at the origin and four substituents at the vertices of an ideal regular tetrahedron:These dimensionless coordinates describe shape, not measured bond lengths. For distinct labels A, B, C and D, define a signed determinantIt is six times the oriented volume of the labelled tetrahedron. A translation cancels from the differences. A rotation $Q$ has $\det Q=+1$ and preserves $\Delta$, whereas the mirror matrix $S=\mathrm{diag}(-1,1,1)$ has $\det S=-1$ and changes it to $-16$. Consequently no rotation can place every distinct label on its mirror partner. This argument uses three-dimensional coordinates, not clockwise order on a page.

Now let C and D both be the same substituent H. Swapping these identical groups is allowed. For this model the half-turnabout the axis $(0,1,1)$ maps the reflected A and B to their original positions and exchanges the two H positions. The models superpose. With distinct C and D, the same half-turn fails precisely at those two labels; the determinant argument rules out every other rotation as well. This is the local reason two identical hydrogen chemical atoms prevent glycine's amino acid alpha carbon from being a tetrahedral stereocenter.

For a flexible molecule, also distinguish conformational isomerism from a persistent enantiomer pair. A particular frozen shape may be chiral while ordinary internal motions interconvert mirror shapes. Turning the entire model is always allowed in a spatial comparison; an internal motion must be considered with its actual accessibility and timescale.