The tomahawk is a tool in

geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...

for angle trisection, the problem of splitting an

angle In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the '' vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles a ...

into three equal parts. The boundaries of its shape include a

semicircle In mathematics (and more specifically geometry), a semicircle is a one-dimensional locus of points that forms half of a circle. The full arc of a semicircle always measures 180° (equivalently, radians, or a half-turn). It has only one line o ...

and two line segments, arranged in a way that resembles a

tomahawk A tomahawk is a type of single-handed axe used by the many Indigenous peoples and nations of North America. It traditionally resembles a hatchet with a straight shaft. In pre-colonial times the head was made of stone, bone, or antler, and Eur ...

, a Native American axe. The same tool has also been called the shoemaker's knife,. but that name is more commonly used in geometry to refer to a different shape, the

arbelos In geometry, an arbelos is a plane region bounded by three semicircles with three apexes such that each corner of each semicircle is shared with one of the others (connected), all on the same side of a straight line (the ''baseline'') that conta ...

(a curvilinear triangle bounded by three mutually tangent semicircles).

Description

The basic shape of a tomahawk consists of a semicircle (the "blade" of the tomahawk), with a line segment the length of the radius extending along the same line as the diameter of the semicircle (the tip of which is the "spike" of the tomahawk), and with another line segment of arbitrary length (the "handle" of the tomahawk) perpendicular to the diameter. In order to make it into a physical tool, its handle and spike may be thickened, as long as the line segment along the handle continues to be part of the boundary of the shape. Unlike a related trisection using a carpenter's square, the other side of the thickened handle does not need to be made parallel to this line segment.. In some sources a full circle rather than a semicircle is used, or the tomahawk is also thickened along the diameter of its semicircle, but these modifications make no difference to the action of the tomahawk as a trisector.

Trisection

To use the tomahawk to

trisect an angle Angle trisection is a classical problem of straightedge and compass construction of ancient Greek mathematics. It concerns construction of an angle equal to one third of a given arbitrary angle, using only two tools: an unmarked straightedge a ...

, it is placed with its handle line touching the apex of the angle, with the blade inside the angle, tangent to one of the two rays forming the angle, and with the spike touching the other ray of the angle. One of the two trisecting lines then lies on the handle segment, and the other passes through the center point of the semicircle.. If the angle to be trisected is too sharp relative to the length of the tomahawk's handle, it may not be possible to fit the tomahawk into the angle in this way, but this difficulty may be worked around by repeatedly doubling the angle until it is large enough for the tomahawk to trisect it, and then repeatedly bisecting the trisected angle the same number of times as the original angle was doubled.. If the apex of the angle is labeled , the point of tangency of the blade is , the center of the semicircle is , the top of the handle is , and the spike is , then triangles and are both

right triangle A right triangle (American English) or right-angled triangle ( British), or more formally an orthogonal triangle, formerly called a rectangled triangle ( grc, ὀρθόσγωνία, lit=upright angle), is a triangle in which one angle is a right a ...

s with a shared base and equal height, so they are

congruent triangles In geometry, two figures or objects are congruent if they have the same shape and size, or if one has the same shape and size as the mirror image of the other. More formally, two sets of points are called congruent if, and only if, one can be t ...

. Because the sides and of triangle are respectively a tangent and a radius of the semicircle, they are at right angles to each other and is also a right triangle; it has the same

hypotenuse In geometry, a hypotenuse is the longest side of a right-angled triangle, the side opposite the right angle. The length of the hypotenuse can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse e ...

as and the same side lengths , so again it is congruent to the other two triangles, showing that the three angles formed at the apex are equal.. Although the tomahawk may itself be constructed using a compass and straightedge, and may be used to trisect an angle, it does not contradict

Pierre Wantzel Pierre Laurent Wantzel (5 June 1814 in Paris – 21 May 1848 in Paris) was a French mathematician who proved that several ancient geometric problems were impossible to solve using only compass and straightedge. In a paper from 1837, Wantzel pr ...

's 1837 theorem that arbitrary angles cannot be trisected by compass and unmarked straightedge alone. The reason for this is that placing the constructed tomahawk into the required position is a form of neusis that is not allowed in compass and straightedge constructions.

History

The inventor of the tomahawk is unknown, but the earliest references to it come from 19th-century France. It dates back at least as far as 1835, when it appeared in a book by

Claude Lucien Bergery Claude Lucien Bergery (1787–1863) was a French economist and management theorist. He was a founder of scientific management. Life The son of an innkeeper, Bergery was born in Orléans. He was a student at the École Polytechnique which he ente ...

, ''Géométrie appliquée à l'industrie, à l'usage des artistes et des ouvriers'' (3rd edition). Another early publication of the same trisection was made by

Henri Brocard Pierre René Jean Baptiste Henri Brocard (12 May 1845 – 16 January 1922) was a French meteorologist and mathematician, in particular a geometer. His best-known achievement is the invention and discovery of the properties of the Brocard point ...

in 1877; Brocard in turn attributes its invention to an 1863 memoir by French naval officer . incorrectly writes these names as Bricard and Glatin.

References

External links

Trisection using special tools: "Tomahawk"
Takaya Iwamoto, 2006, featuring a tomahawk tool made from transparent vinyl and comparisons for accuracy against other trisectors *{{mathworld, title=Tomahawk, urlname=Tomahawk, mode=cs2
Construction heptagon with tomahawk, animation
Mathematical tools