peth₁-
“to spread out, extend”Source of Latin pandere, English expand, patent, petal
Root for spreading or extending, yielding Latin pandere, patēre and English patent, expand, petal.
Discussion
The Proto-Indo-European root *peth₁- meant "to spread out, to extend, to open wide" and produced descendants relating to breadth, openness, and the act of stretching one's arms. The laryngeal *h₁ is reflected in the vowel length of certain derivatives. The root should be distinguished from the similar *pleth₂- "to spread flat", though the two roots share obvious semantic territory.
In Germanic, the initial *p became *f by Grimm's Law. The most notable descendant is fathom, from Old English fæþm "an embrace, the span of outstretched arms" — a unit of measurement (six feet) based on the distance from fingertip to fingertip with arms extended. The word perfectly encodes the root meaning of spreading out. The verb to fathom, meaning "to probe the depth" (originally by lowering a weighted line measured in fathoms), developed its figurative sense of "to understand fully" — to get to the bottom of something.
Latin patēre "to lie open, to be exposed" is the key learned reflex, producing English patent (originally "lying open", hence an open letter granting a right, and later an exclusive right to an invention), paternal in one usage, and the word patio (an open courtyard). The Latin adjective patulus "spreading, open" is related. The word pace (as in the distance covered by one spread-out step) may also connect to this root.
Greek petannynai "to spread out" is a cognate, and the word petalon "a leaf" (something spread flat) gave English petal.
The semantic journey from "to spread out" through "an embrace" (outstretched arms) to "a unit of measurement" to "to understand deeply" is a characteristic example of how concrete physical actions in Proto-Indo-European could evolve, step by logical step, into abstract intellectual vocabulary. The fathom that measures the sea and the fathom that measures understanding are the same word, and both trace back to the simple gesture of opening one's arms.