Bisects and splits refer to postage stamp

Postage stamp

A postage stamp is a small piece of paper that is purchased and displayed on an item of mail as evidence of payment of postage. Typically, stamps are made from special paper, with a national designation and denomination on the face, and a gum adhesive on the reverse side...

s that have been cut in part, most commonly in half, but also other fractions, and postally used for the proportionate value of the entire stamp, such as a two cent stamp cut in half and used as a one cent stamp.
When stocks of a certain stamp

Postage stamp

A postage stamp is a small piece of paper that is purchased and displayed on an item of mail as evidence of payment of postage. Typically, stamps are made from special paper, with a national designation and denomination on the face, and a gum adhesive on the reverse side...

 ran out, postmasters sometimes resorted to cutting higher denominated stamps in half, vertically or diagonally, thus obtaining two "stamps" each representing half of the original monetary value, or "face" value, of the uncut stamp. The general public also resorted to this practice, sometimes pursuant to official or tacit permission and sometimes without any express authorization.

Many of these instances have been well documented in postal history. One example is the bisects of the Island of Guernsey

Guernsey

Guernsey, officially the Bailiwick of Guernsey is a British Crown dependency in the English Channel off the coast of Normandy.The Bailiwick, as a governing entity, embraces not only all 10 parishes on the Island of Guernsey, but also the islands of Herm, Jethou, Burhou, and Lihou and their islet...

 during the German military occupation of the Channel Islands

Channel Islands

The Channel Islands are an archipelago of British Crown Dependencies in the English Channel, off the French coast of Normandy. They include two separate bailiwicks: the Bailiwick of Guernsey and the Bailiwick of Jersey...

 during World War II

World War II

World War II, or the Second World War , was a global conflict lasting from 1939 to 1945, involving most of the world's nations—including all of the great powers—eventually forming two opposing military alliances: the Allies and the Axis...

. Early Mexican stamps are known to have been used cut in half, three-quarters, quarters and even eighths.

Many bisects and splits are considerably more valuable than the stamps from which they were made. These, however, only have philatelic value when the cut portion is still affixed to the envelope or a piece showing the postmarks, as otherwise it cannot be confirmed that the stamp was in fact postally used as a split and not simply a complete stamp that was cut up after use.