Kmhkmh: ←Created page with 'File:Gnomon theorem.svg|thumb|upright=1.3|gnomon: <math> ABFEGD</math> <br/> theoref of the Gnomon: green area = red area, <br/><math>|AHGD|=|ABFI|,\, |HBFE|=|...'
[[File:Gnomon theorem.svg|thumb|upright=1.3|gnomon: <math> ABFEGD</math> <br/> theoref of the Gnomon: green area = red area, <br/><math>|AHGD|=|ABFI|,\, |HBFE|=|IEGD| </math> ]]
[[File:Gnomon division.svg|thumb|upright=1.3|geometrical representation of a division]]
[[File:Gnomon streckenteilung.svg|thumb|upright=1.3|Transferring the ratio of a partition of line segment AB to line segment HG <math>\tfrac{|AH|}{|HB|}=\tfrac{|HE|}{|EG|}</math>]]
The '''theorem of the gnomon''' states that certain [[parallelogram]]s occurring in a [[Gnomon (figure)|gnomon]] have areas of equal size.
== Theorem ==
In a parallelogram <math>ABCD</math> with a point <math>E</math> on the diagonal <math>AC</math> the parallel to <math>AD</math> through <math>E</math> intersects the side <math>CD</math> in <math>G</math> and the side <math>AB</math> in <math>H</math>. Similarly the parallel to the side <math>AB</math> through <math>E</math> intersects the side <math>AD</math> in <math>I</math> and the side <math>BC</math> in <math>F</math>. The theorem of the gnomon now states, that the parallelograms <math>HBFE</math> und <math>IEGD</math> have equal areas..<ref name="HNL">Lorenz Halbeisen, Norbert Hungerbühler, Juan Läuchli: ''Mit harmonischen Verhältnissen zu Kegelschnitten: Perlen der klassischen Geometrie''. Springer 2016, ISBN 9783662530344, pp. 190-191 </ref><ref name="Hazard">William J. Hazard: ''Generalizations of the Theorem of Pythagoras and Euclid's Theorem of the Gnomon''. The American Mathematical Monthly, volume 36, no. 1 (Jan., 1929), pp. 32-34 ([http://bit.ly/2S0fvMn JSTOR]) </ref>
''Gnomon'' is the name for the L-shaped figure consisting of the two overlapping parallelograms <math>ABFI</math> and <math>AHGD</math>. The parallelograms of equal area <math>HBFE</math> und <math>IEGD</math> are called ''complements'' (of the parallelograms on diagonal <math>EFCG</math> und <math>AHEI</math>).<ref name="Tropfke">[[Johannes Tropfke]]: ''Geschichte der Elementarmathematik Ebene Geometrie - Band 4: Ebene Geometrie''. Walter de Gruyter, 2011, ISBN 9783111626932, pp. [http://bit.ly/2CVRfRf 134-135] (German)</ref>
== Applications and extensions ==
The theorem of the gnomon can be used to construct a new parallelogram or rectangle of equal area to a given parallelogram or rectangle by the means of [[straightedge and compass construction]]s. This also allows for the representation of a division of two numbers in geometrical terms, an important feature to reformulate geometrical problems in algebraic terms. More precisely if two numbers are given as lengths of line segments one can construct a third line segment, the length of which matches the quotient of those two numbers (see drawing). Another application is to transfer the ratio of partition of one line segment to another line segment (of different length), that is dividing that other line segment in the same ratio as a given line segment and its partition (see drawing).<ref name="HNL"/>
[[File:Gnomon3d 1.png|thumb|upright=1.3|<math>\mathbb{A}</math> is the (lower) parallepiped around the diagonal with <math>P</math> and its complements <math>\mathbb{B}</math>, <math>\mathbb{C}</math> and <math>\mathbb{D}</math> have the same volume: <math>|\mathbb{B}|=|\mathbb{C}|=|\mathbb{D}|</math>]]
A similar statement can be made in three dimensions for [[parallelepiped]]s. In this case you have a point <math>P</math> on the space diagonal of a parallelepiped and instead of two parallel lines you have three planes through <math>P</math> being parallel to the faces of the parallelepiped. The three planes partition the parallelepiped into eight smaller parallelepipeds, two of those surround the diagonal and meet in <math>P</math>. Now each of those two parallepipeds around the diagonal has three of the remaining six parallelepipeds attached to it and those three play the role of the complements and are of equal volume (see drawing).<ref name="Hazard"/>
== Historical aspects ==
The theorem of the gnomon is already described in [[Euclid's Elements]] (ca 300 BC) and there it plays an important role in the derivation of other theorems. It is given as proposition 43 in the first book of the Elements, where it is phrased as a statement about parallelograms without using the term gnomon. The latter is introduced by Euclid as the second definition of the second book of Elements. Further theorems for which the gnomon and its properties play an important role are proposition 6 in book II, proposition 29 in book VI and the propositionen 1, 2, 3 und 4 in book XIII.<ref name="VA">Paolo Vighi, Igino Aschieri: ''From Art to Mathematics in the Paintings of Theo van Doesburg''. In: Vittorio Capecchi (Hrsg.), Massimo Buscema (Hrsg.), Pierluigi Contucci (Hrsg.), Bruno D'Amore (Hrsg.): ''Applications of Mathematics in Models, Artificial Neural Networks and Arts''. Springer, 2010, ISBN 9789048185818, pp. 601-610, in particular pp. 603-606 </ref><ref name"Fischler">Roger Herz-Fischler: ''A Mathematical History of the Golden Number''. Dover, 2013, ISBN 9780486152325, pp.[http://bit.ly/2RVfAAO 35-36] </ref><ref name="Evans">George W. Evans: ''Some of Euclid's Algebra''. The Mathematics Teacher, Volume 20, no. 3 (March, 1927), pp. 127-141 ([http://bit.ly/2CY7RYi JSTOR]) </ref>
== References ==
*Lorenz Halbeisen, Norbert Hungerbühler, Juan Läuchli: ''Mit harmonischen Verhältnissen zu Kegelschnitten: Perlen der klassischen Geometrie''. Springer 2016, ISBN 9783662530344, pp. 190–191 (German)
*George W. Evans: ''Some of Euclid's Algebra''. The Mathematics Teacher, Band 20, Nr. 3 (März, 1927), pp. 127–141 ([http://bit.ly/2CY7RYi JSTOR])
*William J. Hazard: ''Generalizations of the Theorem of Pythagoras and Euclid's Theorem of the Gnomon''. The American Mathematical Monthly, Band 36, Nr. 1 (Jan., 1929), pp. 32–34 ([http://bit.ly/2S0fvMn JSTOR])
*Paolo Vighi, Igino Aschieri: ''From Art to Mathematics in the Paintings of Theo van Doesburg''. In: Vittorio Capecchi (Hrsg.), Massimo Buscema (Hrsg.), Pierluigi Contucci (Hrsg.), Bruno D'Amore (Hrsg.): ''Applications of Mathematics in Models, Artificial Neural Networks and Arts''. Springer, 2010, ISBN 9789048185818, pp. 601–610
== External links ==
* [http://bit.ly/2RU0ZWe ''theorem of the gnomon''] and [http://bit.ly/2CWjwa4 ''definition des Gnomons''] in Euclid's Elementen
== Notes ==
<references/>
[[Category:Euclidean geometry]]
[[Category:Theorems in geometry]]
[[File:Gnomon division.svg|thumb|upright=1.3|geometrical representation of a division]]
[[File:Gnomon streckenteilung.svg|thumb|upright=1.3|Transferring the ratio of a partition of line segment AB to line segment HG <math>\tfrac{|AH|}{|HB|}=\tfrac{|HE|}{|EG|}</math>]]
The '''theorem of the gnomon''' states that certain [[parallelogram]]s occurring in a [[Gnomon (figure)|gnomon]] have areas of equal size.
== Theorem ==
In a parallelogram <math>ABCD</math> with a point <math>E</math> on the diagonal <math>AC</math> the parallel to <math>AD</math> through <math>E</math> intersects the side <math>CD</math> in <math>G</math> and the side <math>AB</math> in <math>H</math>. Similarly the parallel to the side <math>AB</math> through <math>E</math> intersects the side <math>AD</math> in <math>I</math> and the side <math>BC</math> in <math>F</math>. The theorem of the gnomon now states, that the parallelograms <math>HBFE</math> und <math>IEGD</math> have equal areas..<ref name="HNL">Lorenz Halbeisen, Norbert Hungerbühler, Juan Läuchli: ''Mit harmonischen Verhältnissen zu Kegelschnitten: Perlen der klassischen Geometrie''. Springer 2016, ISBN 9783662530344, pp. 190-191 </ref><ref name="Hazard">William J. Hazard: ''Generalizations of the Theorem of Pythagoras and Euclid's Theorem of the Gnomon''. The American Mathematical Monthly, volume 36, no. 1 (Jan., 1929), pp. 32-34 ([http://bit.ly/2S0fvMn JSTOR]) </ref>
''Gnomon'' is the name for the L-shaped figure consisting of the two overlapping parallelograms <math>ABFI</math> and <math>AHGD</math>. The parallelograms of equal area <math>HBFE</math> und <math>IEGD</math> are called ''complements'' (of the parallelograms on diagonal <math>EFCG</math> und <math>AHEI</math>).<ref name="Tropfke">[[Johannes Tropfke]]: ''Geschichte der Elementarmathematik Ebene Geometrie - Band 4: Ebene Geometrie''. Walter de Gruyter, 2011, ISBN 9783111626932, pp. [http://bit.ly/2CVRfRf 134-135] (German)</ref>
== Applications and extensions ==
The theorem of the gnomon can be used to construct a new parallelogram or rectangle of equal area to a given parallelogram or rectangle by the means of [[straightedge and compass construction]]s. This also allows for the representation of a division of two numbers in geometrical terms, an important feature to reformulate geometrical problems in algebraic terms. More precisely if two numbers are given as lengths of line segments one can construct a third line segment, the length of which matches the quotient of those two numbers (see drawing). Another application is to transfer the ratio of partition of one line segment to another line segment (of different length), that is dividing that other line segment in the same ratio as a given line segment and its partition (see drawing).<ref name="HNL"/>
[[File:Gnomon3d 1.png|thumb|upright=1.3|<math>\mathbb{A}</math> is the (lower) parallepiped around the diagonal with <math>P</math> and its complements <math>\mathbb{B}</math>, <math>\mathbb{C}</math> and <math>\mathbb{D}</math> have the same volume: <math>|\mathbb{B}|=|\mathbb{C}|=|\mathbb{D}|</math>]]
A similar statement can be made in three dimensions for [[parallelepiped]]s. In this case you have a point <math>P</math> on the space diagonal of a parallelepiped and instead of two parallel lines you have three planes through <math>P</math> being parallel to the faces of the parallelepiped. The three planes partition the parallelepiped into eight smaller parallelepipeds, two of those surround the diagonal and meet in <math>P</math>. Now each of those two parallepipeds around the diagonal has three of the remaining six parallelepipeds attached to it and those three play the role of the complements and are of equal volume (see drawing).<ref name="Hazard"/>
== Historical aspects ==
The theorem of the gnomon is already described in [[Euclid's Elements]] (ca 300 BC) and there it plays an important role in the derivation of other theorems. It is given as proposition 43 in the first book of the Elements, where it is phrased as a statement about parallelograms without using the term gnomon. The latter is introduced by Euclid as the second definition of the second book of Elements. Further theorems for which the gnomon and its properties play an important role are proposition 6 in book II, proposition 29 in book VI and the propositionen 1, 2, 3 und 4 in book XIII.<ref name="VA">Paolo Vighi, Igino Aschieri: ''From Art to Mathematics in the Paintings of Theo van Doesburg''. In: Vittorio Capecchi (Hrsg.), Massimo Buscema (Hrsg.), Pierluigi Contucci (Hrsg.), Bruno D'Amore (Hrsg.): ''Applications of Mathematics in Models, Artificial Neural Networks and Arts''. Springer, 2010, ISBN 9789048185818, pp. 601-610, in particular pp. 603-606 </ref><ref name"Fischler">Roger Herz-Fischler: ''A Mathematical History of the Golden Number''. Dover, 2013, ISBN 9780486152325, pp.[http://bit.ly/2RVfAAO 35-36] </ref><ref name="Evans">George W. Evans: ''Some of Euclid's Algebra''. The Mathematics Teacher, Volume 20, no. 3 (March, 1927), pp. 127-141 ([http://bit.ly/2CY7RYi JSTOR]) </ref>
== References ==
*Lorenz Halbeisen, Norbert Hungerbühler, Juan Läuchli: ''Mit harmonischen Verhältnissen zu Kegelschnitten: Perlen der klassischen Geometrie''. Springer 2016, ISBN 9783662530344, pp. 190–191 (German)
*George W. Evans: ''Some of Euclid's Algebra''. The Mathematics Teacher, Band 20, Nr. 3 (März, 1927), pp. 127–141 ([http://bit.ly/2CY7RYi JSTOR])
*William J. Hazard: ''Generalizations of the Theorem of Pythagoras and Euclid's Theorem of the Gnomon''. The American Mathematical Monthly, Band 36, Nr. 1 (Jan., 1929), pp. 32–34 ([http://bit.ly/2S0fvMn JSTOR])
*Paolo Vighi, Igino Aschieri: ''From Art to Mathematics in the Paintings of Theo van Doesburg''. In: Vittorio Capecchi (Hrsg.), Massimo Buscema (Hrsg.), Pierluigi Contucci (Hrsg.), Bruno D'Amore (Hrsg.): ''Applications of Mathematics in Models, Artificial Neural Networks and Arts''. Springer, 2010, ISBN 9789048185818, pp. 601–610
== External links ==
* [http://bit.ly/2RU0ZWe ''theorem of the gnomon''] and [http://bit.ly/2CWjwa4 ''definition des Gnomons''] in Euclid's Elementen
== Notes ==
<references/>
[[Category:Euclidean geometry]]
[[Category:Theorems in geometry]]
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