A cube is a hexahedron of six identical squares. Duplication of a cube; or the Delian Problem, means starting with a cube of edge e, having a volume if v3 then proceeding to replace it by another cube of edge k, having volume, 2e3. To construct the replacement cube requires the construction of k = e 3√2. Not until the 19thCentury was it proved that there was no possible geometric construction for k= e3√ 2. The solution to this problem is to bypass the impossible and deal with the possible by starting with a range of cubes of exactly known edges, en, and their corresponding exactly known volumes, Vn, and then establishing graphically the relationship between en and Vn, to produce a practical tool. Using this practical tool in conjunction with the well established problem solving technique of working backwards, any exact value of e and its corresponding value of volume V are determined by compass extent. A determination is made of the volume of the replacement cube- the duplication- by repeating in adjoining sequence, this compass extent twice on the volume axis. Identification of the corresponding value of edge e, on the graph gives the value of k, the edge of the replacement cube. With compass extent of value, k, the cube that duplicates the original is constructed.
| Published in | American Journal of Applied Mathematics (Volume 3, Issue 6) |
| DOI | 10.11648/j.ajam.20150306.13 |
| Page(s) | 256-258 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2024. Published by Science Publishing Group |
Exact, Practical Tool, Working Backwards
| [1] | “Duplication of a Cube” (2014, August 23) retrieved from http://www.cut-the-not.org/do_you_know/Delian.shtml |
| [2] | Tapson, Frank, “Oxford Mathematics Study Dictionary”, p.46, Oxford University Press, 2006. |
| [3] | “Doubling the Cube” (2014, August 28) retrieved from http://encyclopedia2.thefreedictionary.com/Cube+(geometry) |
| [4] | ‘The Delian Problem” (2014, August 28) retrieved from http://www.takayaiwamoto.com/Greek_Math/Delian/History_Delian.html |
| [5] | “Doubling the cube” (2014, August 30) retrieved from http://en.wikipedia.org/wiki/Doubling_the_cube |
| [6] | Claphan, Christopher, “The concise Oxford Dictionary of Mathematics Oxford Reference”, p. 51 Oxford University Press, New York, 1990. |
| [7] | Edited by Iyanaga, Shokichi; Kawada, Yukiyosi (translated by Mathematical Society of Japan, American Mathematical Society) “Encyclopedic Dictionary of Mathematics: Mathematical Society of Japan” p. 588 MIT press, Cambridge 1968. |
| [8] | “Duplicating a Cube” (2014, September 6) retrieved from http://www.amsi.org.au/teacher_modules/Construction.html |
| [9] | “Duplicating a Cube” (2014, November 17) retrieved from http://mathandmultimedia.com/2013/07/07/3-ancient-construction-problems/ |
| [10] | “Duplication of a Cube” (2014, November 19) p.28 retrieved from http://www-math.ucdenver.edu/~wcherowi/courses/m3210/lecchap5.pdf |
| [11] | New Encyclopedia Britannica, Vol 23 Macropedia-Knowledge in Depth. Founded 1768 15th Edition, pp.581, 583 Inc. Jacob E. Safra, Chairman of the board Jorge Aguillar-Cuiz, President. London/New Delhi/Paris/Seoul/Sydney/Taipei/Tokyo © 2005. |
| [12] | Boyer, Carl B, Collier’s Encyclopedia, Vol 7 p. 160 Maxwell Macmillan Publishing Group, Copyright © 1992. |
| [13] | Rosenberg, Alex, Collier’s Encyclopedia, Vol 1 p. 543 Maxwell Macmillan Publishing Group, Copyright © 1992. |
APA Style
Lorna A. Willis. (2015). Duplication of a Cube. American Journal of Applied Mathematics, 3(6), 256-258. https://doi.org/10.11648/j.ajam.20150306.13
ACS Style
Lorna A. Willis. Duplication of a Cube. Am. J. Appl. Math. 2015, 3(6), 256-258. doi: 10.11648/j.ajam.20150306.13
AMA Style
Lorna A. Willis. Duplication of a Cube. Am J Appl Math. 2015;3(6):256-258. doi: 10.11648/j.ajam.20150306.13
Share This Article
Twitter
Linked In
Facebook
Copy
Download@article{10.11648/j.ajam.20150306.13,
author = {Lorna A. Willis},
title = {Duplication of a Cube},
journal = {American Journal of Applied Mathematics},
volume = {3},
number = {6},
pages = {256-258},
doi = {10.11648/j.ajam.20150306.13},
url = {https://doi.org/10.11648/j.ajam.20150306.13},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20150306.13},
abstract = {A cube is a hexahedron of six identical squares. Duplication of a cube; or the Delian Problem, means starting with a cube of edge e, having a volume if v3 then proceeding to replace it by another cube of edge k, having volume, 2e3. To construct the replacement cube requires the construction of k = e 3√2. Not until the 19thCentury was it proved that there was no possible geometric construction for k= e3√ 2. The solution to this problem is to bypass the impossible and deal with the possible by starting with a range of cubes of exactly known edges, en, and their corresponding exactly known volumes, Vn, and then establishing graphically the relationship between en and Vn, to produce a practical tool. Using this practical tool in conjunction with the well established problem solving technique of working backwards, any exact value of e and its corresponding value of volume V are determined by compass extent. A determination is made of the volume of the replacement cube- the duplication- by repeating in adjoining sequence, this compass extent twice on the volume axis. Identification of the corresponding value of edge e, on the graph gives the value of k, the edge of the replacement cube. With compass extent of value, k, the cube that duplicates the original is constructed.},
year = {2015}
}
TY - JOUR T1 - Duplication of a Cube AU - Lorna A. Willis Y1 - 2015/10/27 PY - 2015 N1 - https://doi.org/10.11648/j.ajam.20150306.13 DO - 10.11648/j.ajam.20150306.13 T2 - American Journal of Applied Mathematics JF - American Journal of Applied Mathematics JO - American Journal of Applied Mathematics SP - 256 EP - 258 PB - Science Publishing Group SN - 2330-006X UR - https://doi.org/10.11648/j.ajam.20150306.13 AB - A cube is a hexahedron of six identical squares. Duplication of a cube; or the Delian Problem, means starting with a cube of edge e, having a volume if v3 then proceeding to replace it by another cube of edge k, having volume, 2e3. To construct the replacement cube requires the construction of k = e 3√2. Not until the 19thCentury was it proved that there was no possible geometric construction for k= e3√ 2. The solution to this problem is to bypass the impossible and deal with the possible by starting with a range of cubes of exactly known edges, en, and their corresponding exactly known volumes, Vn, and then establishing graphically the relationship between en and Vn, to produce a practical tool. Using this practical tool in conjunction with the well established problem solving technique of working backwards, any exact value of e and its corresponding value of volume V are determined by compass extent. A determination is made of the volume of the replacement cube- the duplication- by repeating in adjoining sequence, this compass extent twice on the volume axis. Identification of the corresponding value of edge e, on the graph gives the value of k, the edge of the replacement cube. With compass extent of value, k, the cube that duplicates the original is constructed. VL - 3 IS - 6 ER -
Information
Email: