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I know that any carbon to show chirality has to be bonded to 4 different atoms (groups of atoms). So a double bond is considered as that the carbon is achiral. I learnt from my lecturer that dissubstituted allene is an exception. If it is true that sp2 hybridised carbon cannot show chirality, are there any other exceptions?

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    $\begingroup$ Hexahelicene (and similar molecules) consist entirely of sp2 carbons but are chiral because of steric effects that constrain their overall shape. $\endgroup$ – matt_black Dec 19 '18 at 16:50
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    $\begingroup$ Then strictly we do not have sp2 hybridization. You have to allow some latitude in the interpretation, $\endgroup$ – Oscar Lanzi Dec 19 '18 at 17:48
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    $\begingroup$ Carbons (or any other atoms) don't show chirality; molecules do. $\endgroup$ – Ivan Neretin Dec 19 '18 at 18:37
  • $\begingroup$ Ivan Neretin then what about chirality centres $\endgroup$ – Adhesh Sagar Dec 20 '18 at 0:10
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    $\begingroup$ Chirality is not necessarily produced by chiral centers. $\endgroup$ – Ivan Neretin Dec 20 '18 at 10:29
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You are confusing two different concepts—and I don’t blame you because many chemists don’t cleanly separate the two concepts.

One concept is that of an asymmetric (carbon) atom. To establish whether a carbon atom is asymmetric, compare it to its mirror image. If you can overlap perfectly only by rotation and translation, the atom is symmetric; if the two mirror images act as left and right hands, it is asymmetric. Since $\mathrm{sp^2}$ carbon atoms are, by definition, in a planar environment, they will always be symmetric because a plane of symmetry will always traverse them.

The second concept is overall chirality of a molecule. For this, you basically do the same thing as you did for an individual carbon except you mirror the entire molecule and attempt to overlap it with the original molecule. This, of course, depends on the overall 3D structure. Again, if the molecule as a whole contains a plane of symmetry, it is by definition achiral.

The two definitions of asymmetry are nowhere near equivalent. It is possible for a compound to be composed entirely of asymmetric carbon atoms and yet be achiral, e.g. meso-1,2-dichloro-1,2-difluoroethane. Likewise, it is common for a molecule that consists entirely of symmetric carbon atoms to be overall asymmetric like BINOL, allenes or helicenes. Thus, one should always carefully separate the two concepts to not accidentally cause a mix-up.

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As correctly stated in a comment by @Ivan Neretin, chirality is not bound to stereogenic centers (e.g. the textbook examples of carbon atoms with four different substitutents, or chiral sulfoxides) only.

Returning to your question, an other option to build chirality is of course axial chirality, where the allenes -- formally built from $sp^2$-hybridized carbon atoms -- offer enantiomeric forms.

enter image description here (source)

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