116:, 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
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as meaning "a family of constructions dependent upon a single parameter" in which, as the parameter varies, some combinatorial change in the construction occurs at the desired parameter value. La Nave and Mazur describe other trisections than the tomahawk, but the same description applies here: a
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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
72:"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
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tomahawk placed with its handle on the apex, parameterized by the position of the spike on its ray, gives a family of constructions in which the relative positions of the blade and its ray change as the spike is placed at the correct point.
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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.
227:'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
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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
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Mathematical
Carnival: from penny puzzles, card shuffles and tricks of lightning calculators to roller coaster rides into the fourth dimension
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690:, Takaya Iwamoto, 2006, featuring a tomahawk tool made from transparent vinyl and comparisons for accuracy against other trisectors
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500:"Recherches sur les moyens de reconnaître si un Problème de Géométrie peut se résoudre avec la règle et le compas"
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384:, Dolciani Mathematical Expositions, vol. 42, Mathematical Association of America, pp. 147–148,
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are respectively a tangent and a radius of the semicircle, they are at right angles to each other and
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in 1877; Brocard in turn attributes its invention to an 1863 memoir by French naval officer
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580:, New Mathematical Library, vol. 13, Mathematical Association of America, p. 87,
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Alsina, Claudi; Nelsen, Roger B. (2010), "9.4 The shoemaker's knife and the salt cellar",
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Yates, Robert C. (1941), "The
Trisection Problem, Chapter III: Mechanical trisectors",
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the trisected angle the same number of times as the original angle was doubled.
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247:(3rd edition). Another early publication of the same trisection was made by
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Géométrie appliquée à l'industrie, à l'usage des artistes et des ouvriers
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63:(a curvilinear triangle bounded by three mutually tangent semicircles).
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Mémoires de la Société des
Sciences physiques et naturelles de Bordeaux
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630:"De quelques moyens pratiques de diviser les angles en parties Ă©gales"
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that is not allowed in compass and straightedge constructions.
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into three equal parts. The boundaries of its shape include a
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Although the tomahawk may itself be constructed using a
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incorrectly writes these names as
Bricard and Glatin.
382:
Charming Proofs: A Journey Into
Elegant Mathematics
161:with a shared base and equal height, so they are
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23:A tomahawk, with its handle and spike thickened
713:Construction heptagon with tomahawk, animation
577:Episodes from the Early History of Mathematics
474:, Jones & Bartlett Learning, p. 191,
610:Bulletin de la Société Mathématique de France
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504:Journal de Mathématiques Pures et Appliquées
606:"Note sur la division mécanique de l'angle"
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688:Trisection using special tools: "Tomahawk"
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411:Fundamental Concepts of Algebra
529:The Mathematical Intelligencer
39:, the problem of splitting an
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442:Geometry for College Students
289:National Mathematics Magazine
527:(2002), "Reading Bombelli",
135:, the top of the handle is
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440:Isaacs, I. Martin (2009),
408:Meserve, Bruce E. (1982),
202:and the same side lengths
16:Tool for trisecting angles
652:George E. Martin (1998),
340:, Knopf, pp. 262–263
221:compass and straightedge
659:Geometric Constructions
112:To use the tomahawk to
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466:Eves, Howard Whitley
253:Pierre-Joseph Glotin
165:. Because the sides
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163:congruent triangles
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94:trisecting an angle
728:Mathematical tools
696:Weisstein, Eric W.
541:10.1007/BF03025306
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74:carpenter's square
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353:Dudley, Underwood
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602:Brocard, H.
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256: [
92:A tomahawk
67:Description
700:"Tomahawk"
662:, Springer
266:References
193:hypotenuse
84:Trisection
45:semicircle
705:MathWorld
654:"Preface"
640:: 253–278
557:189888034
157:are both
118:bisecting
722:Category
604:(1877),
574:(1997),
498:(1837),
468:(1995),
355:(1996),
336:(1975),
53:tomahawk
47:and two
33:geometry
29:tomahawk
616:: 43–47
549:1889932
317:1569903
309:3028413
235:History
61:arbelos
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229:neusis
553:S2CID
305:JSTOR
260:]
41:angle
582:ISBN
476:ISBN
446:ISBN
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386:ISBN
361:ISBN
171:and
150:and
35:for
27:The
537:doi
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199:ACD
195:as
188:ABC
181:ABC
154:ADE
147:ACD
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141:E
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