函

Also used in mathematical and formal senses such as “function (函數).”

Pictograph:

Originally a drawing of a quiver or box used to store arrows — a long, narrow container with an opening at the top.

In oracle bone script (甲骨文) and bronze inscriptions (金文 jinwen), 函 depicted a rectangular container, often shown with an arched lid, representing an enclosing vessel.

From this literal meaning of “to contain, to enclose”, it came to signify a box or case in general, and later anything that contains or stores — whether physical (like a chest) or abstract (like a “function” containing a rule).

서한 (書函) — a letter or correspondence

냉동함 (冷凍函) — freezer compartment

보관함 (保管函) — storage box

함수 (函數) — mathematical function

우편함 (郵便函) — postbox, mailbox

In modern Korean, 函 primarily appears in technical or formal compounds.

It retains both the concrete sense of container and the abstract sense of function, enclosure, or correspondence.

Notably, in Japan, 函 was excluded from the Tōyō Kanji (当用漢字) list, leading to replacement by 関 (관) in many technical words.

Thus, function is written 関数 (かんすう) instead of 函數, though the original reading remains visible in place names such as 函館 (はこだて, Hakodate).

Classical & historical notes:

In ancient Chinese usage, 函 signified:

A container or case — particularly for storing weapons or letters.

「置書於函中。」

“Place the letter inside the box.” — Han dynasty epistolary usage

A narrow valley or pass, by metaphorical extension (as in 函谷關, “Hangu Pass”), referring to a mountain passage enclosed like a box.

By the Han dynasty, 函 had extended meanings such as “enclose, contain,” leading to metaphorical uses in scholarly and mathematical contexts.

In mathematics and logic, 函數 (function) literally means “a containing number or rule,” reflecting the sense of a “container of relationships.”

Cultural & historical notes:

Literal sense — “a container”:

函 originally depicted a case or quiver, symbolizing the concept of containment and protection.

This sense survives in everyday Korean words like 보관함 (locker) or 금고함 (safe box), and in compounds such as 우편함 (mailbox).

Toponymic sense — “enclosed valley”:

In ancient geography, 函 described narrow passes surrounded by mountains, e.g. 函谷關 (Hángǔ Pass), the historical western entrance to the Central Plain.

Abstract sense — “containment of meaning”:

In philosophical and mathematical terminology, 函 symbolizes conceptual encapsulation — something that holds or represents a relationship, hence its adoption in 數學 (mathematics) as “function (函數).”

Symbolic interpretation:

函 embodies the idea of containment, preservation, and enclosure — both material and intellectual.

It represents the boundary that protects or defines content, whether storing arrows, letters, or abstract values.

「函中之物，藏而不露。」

“That which lies within the box — hidden, yet preserved.”

Thus, 函 stands as a symbol of guarded knowledge, communication, and structured containment — from the ancient quiver of the warrior, to the function of the mathematician, to the message sealed in a letter.

Toponymic Note:

函谷關 (Hángǔguān, 함곡관)

One of the most historically and culturally significant uses of 函 appears in the name 函谷關 (Hángǔguān, 함곡관) — the Hangu Pass, an ancient mountain pass on the border between the Guanzhong Plain (關中) and the Henan Basin.

The name literally means “Box-shaped Valley Pass”, from 函 (box, enclosed space) + 谷 (valley) + 關 (gate, pass).

The imagery of 函 here emphasizes enclosure and narrowness, describing a deep, box-like gorge through which travelers had to pass to reach the Central Plain (中原).

In Chinese history, 函谷關 was one of the most strategic gateways:

Built during the Spring and Autumn period, it served as the eastern gate of the Qin state.

It was famously sealed by Laozi (老子) in legend — according to tradition, when he departed westward, the guardian Yin Xi (尹喜) persuaded him to record his wisdom before passing through the gate, resulting in the composition of the Dao De Jing (道德經).

During the Han dynasty, it symbolized the threshold between chaos and order, outer and inner civilization, much like the “frontier” concept in Western history.

Because of this deep symbolism, 函谷關 became a literary metaphor for the border between enlightenment and obscurity, or the gateway to higher understanding.

Later poets used “函谷之門” (“the Gate of Hangu”) to signify both a physical threshold and a spiritual crossing point — a place where one departs from the familiar to encounter the vast unknown.

Later variants include 凾 and 圅, the latter being regarded as the original form of 函.