Etymology
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Words related to uni-

*oi-no- 
Proto-Indo-European root meaning "one, unique."

It forms all or part of: a (1) indefinite article; alone; an; Angus; anon; atone; any; eleven; inch (n.1) "linear measure, one-twelfth of a foot;" lone; lonely; non-; none; null; once; one; ounce (n.1) unit of weight; quincunx; triune; unanimous; unary; une; uni-; Uniate; unilateral; uncial; unicorn; union; unique; unison; unite; unity; universal; universe; university; zollverein.

It is the hypothetical source of/evidence for its existence is provided by: Greek oinos "ace (on dice);" Latin unus "one;" Old Persian aivam; Old Church Slavonic -inu, ino-; Lithuanian vienas; Old Irish oin; Breton un "one;" Old English an, German ein, Gothic ains "one."
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unicameral (adj.)
1853, from uni- "one" + Late Latin camera "chamber" (see camera) + -al (1).
unicycle (n.)
1869, American English, from Latin uni- "one" (see uni-) + -cycle, from bicycle (from Greek kyklos "circle, wheel").
unidirectional (adj.)
1883, from uni- + directional (see direction).
uniform (adj.)

1530s, "of one form," from French uniforme (14c.), from Latin uniformis "having only one form or shape," from uni- "one" (see uni-) + forma "form" (see form (n.)). Related: Uniformly.

unify (v.)

c. 1500, "to make into one," from French unifier (14c.) or directly from Late Latin unificare "make one," from Latin uni- "one" (see uni-) + combining form of facere "to make" (from PIE root *dhe- "to set, put"). Related: Unified; unifying. Unified (field) theory in physics is recorded from 1935.

unipersonal (adj.)
1810, from uni- + personal (adj.).
unipolar (adj.)
1812, originally in electricity, from uni- + polar. Related: Unipolarity.
unisex (adj.)
"sexually indistinguishable or neutral," 1968, from uni- + sex (n.).