{"id":4349,"date":"2022-09-12T15:19:31","date_gmt":"2022-09-12T13:19:31","guid":{"rendered":"https:\/\/erlebnisland-mathematik.de\/?page_id=4349"},"modified":"2023-01-26T14:29:57","modified_gmt":"2023-01-26T13:29:57","slug":"advanced-text-moebius-street","status":"publish","type":"page","link":"https:\/\/erlebnisland-mathematik.de\/en\/advanced-text-moebius-street\/","title":{"rendered":"Advanced text M\u00f6bius Street"},"content":{"rendered":"

[vc_row drowwidth=”sidebar-biest-default sidebar-biest”][vc_column][vc_column_text]<\/p>\n

M\u00f6bius Street<\/h1>\n

The M\u00f6bius strip<\/em> (also: M\u00f6bius loop<\/em> or M\u00f6bius band<\/em>) is a (two-dimensional) surface that has only one edge<\/em> and only one side<\/em>:[\/vc_column_text][vc_single_image image=”1315″ img_size=”large” alignment=”center”]Figure 1: The M\u00f6bius strip[\/vc_single_image][vc_column_text]The M\u00f6bius strip was described independently in 1858 by the G\u00f6ttingen professor of mathematics and physics, Johann Benedict Listing<\/em> (1808–1892), and the astronomer August Ferdinand M\u00f6bius<\/em> (1790–1868). Mathematically, it is a non-orientable manifold.<\/p>\n

The construction of a M\u00f6bius strip can be understood in a simple way if one uses the comparable construction of a “normal<\/em>” ring for comparison: Two long strips (of the same width) with parallel edges are cut out of a sheet of paper. In the first — the comparison strip — the two ends are smoothly joined together (e.g. glued) to form a “normal<\/em>” ring. The second strip, the actual M\u00f6bius strip, is twisted half a turn (180\u00b0) before being joined. In this way, the M\u00f6bius strip becomes a fascinating little toy that can be easily made.<\/p>\n

Now one can observe special phenomena: The ring-shaped twisted band, although it was created in a simple way from one<\/em> strip of paper with an originally square<\/em> edge and a clearly definable lower surface and surface, now has only one<\/em> edge and one<\/em> side.<\/p>\n

Another way of looking at it leads to the conclusion that there is no “inside<\/em>” and no “outside<\/em>” on the M\u00f6bius strip, just as there is no “above<\/em>” and “below<\/em>“.<\/p>\n

The following feature is also noteworthy:<\/p>\n

A M\u00f6bius strip can be cut along its centre line without breaking into two separate rings half as wide as is automatically the case with the “normal<\/em>” ring. The inner twist is then no longer only 180\u00b0, but 360\u00b0. If one repeats this cutting again, the M\u00f6bius strip disintegrates into two separate, intertwined individual strips.<\/p>\n

In the visual arts, there are now famous depictions of the M\u00f6bius strip, e.g. that of the Dutch graphic artist Maurits Cornelis Escher<\/span><\/em> (1963), the “Colossus of Frankfurt<\/em>” by Max Bill<\/em> (1908–1994) and the painting by the Dresden professor of geometry, Gert B\u00e4r<\/em> (“Catastrophe in the M\u00f6bius strip<\/em>“, 1968).<\/p>\n

For illustration purposes, the M\u00f6bius strip in the Maths Adventure Land is driven on by a small vehicle, the M\u00f6biusmobile (see Figure 1) (a technical masterpiece by the company …tronikDesign<\/em>). It is also equipped with a mini camera whose images are continuously transmitted on a monitor. Incidentally, conveyor belts and drive belts are also manufactured as M\u00f6bius belts so that the supposed top or bottom side wears evenly.[\/vc_column_text][vc_single_image image=”1319″ img_size=”large” alignment=”center”]Figure 2: The M\u00f6bius Mobile on the M\u00f6bius strip[\/vc_single_image][vc_single_image image=”1323″ img_size=”large” alignment=”center”]Figure 3: The M\u00f6bius Street[\/vc_single_image][vc_single_image image=”1327″ img_size=”large” alignment=”center”]Figure 4[\/vc_single_image][vc_column_text]<\/p>\n

And now … the mathematics:<\/h3>\n

[\/vc_column_text][vc_column_text]The M\u00f6bius strip is a two-dimensional subset (area) in the three-dimensional space \"\mathbb R^3\", which can be described by the following parameter representation<\/em>: <\/p>\n

  <\/span>   <\/span>\"\[\begin {pmatrix} x\\ y\\ z\end {pmatrix} =\begin {pmatrix} \cos(\alpha)\left(1+\frac{r}{2}\cos(\alpha/2)\right)\\ \sin(\alpha)\left(1+\frac{r}{2}\cos(\alpha/2)\right)\\ \frac{r}{2}\sin(\alpha/2)\end{pmatrix}.\]\"<\/p>\n

Here \"0\leq\alpha\lt 2\pi\" (“orbital parameter<\/em>“) and \"-1\lt r\lt 1\" (“radial parameter<\/em>“) apply.[\/vc_column_text][vc_column_text]<\/p>\n

Literature<\/h3>\n

[\/vc_column_text][vc_column_text][1] Greenland, C.: Begegnungen auf dem M\u00f6biusband<\/em>, Roman, M\u00fcnchen, 1996.<\/p>\n

[2] Herges, R.: M\u00f6bius, Escher, Bach — Das unendliche Band in Kunst und Wissenschaft<\/em>, in: Naturwissenschaftliche Rundschau (6\/58), Darmstadt, 2005.<\/p>\n

[3] Paenza, A.: Mathematik durch die Hintert\u00fcr — Band 2: Vom M\u00f6biusband zum Pascalschen Dreieick — neue spannende Ausfl\u00fcge in die Welt der Zahlen<\/em>, M\u00fcnchen, 2009.[\/vc_column_text][\/vc_column][\/vc_row]<\/p>\n<\/div>","protected":false},"excerpt":{"rendered":"

[vc_row drowwidth=”sidebar-biest-default sidebar-biest”][vc_column][vc_column_text] M\u00f6bius Street The M\u00f6bius strip (also: M\u00f6bius loop or M\u00f6bius band) is a (two-dimensional) surface that has only one edge and only one side:[\/vc_column_text][vc_single_image image=”1315″ img_size=”large” alignment=”center”]Figure 1: The M\u00f6bius strip[\/vc_single_image][vc_column_text]The M\u00f6bius strip was described independently in 1858 by the G\u00f6ttingen professor of mathematics and physics, Johann Benedict Listing (1808–1892), and the … Advanced text M\u00f6bius Street<\/span><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_acf_changed":false,"footnotes":""},"folder":[],"class_list":["post-4349","page","type-page","status-publish","hentry"],"acf":[],"_links":{"self":[{"href":"https:\/\/erlebnisland-mathematik.de\/en\/wp-json\/wp\/v2\/pages\/4349","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/erlebnisland-mathematik.de\/en\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/erlebnisland-mathematik.de\/en\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/erlebnisland-mathematik.de\/en\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/erlebnisland-mathematik.de\/en\/wp-json\/wp\/v2\/comments?post=4349"}],"version-history":[{"count":5,"href":"https:\/\/erlebnisland-mathematik.de\/en\/wp-json\/wp\/v2\/pages\/4349\/revisions"}],"predecessor-version":[{"id":4412,"href":"https:\/\/erlebnisland-mathematik.de\/en\/wp-json\/wp\/v2\/pages\/4349\/revisions\/4412"}],"wp:attachment":[{"href":"https:\/\/erlebnisland-mathematik.de\/en\/wp-json\/wp\/v2\/media?parent=4349"}],"wp:term":[{"taxonomy":"folder","embeddable":true,"href":"https:\/\/erlebnisland-mathematik.de\/en\/wp-json\/wp\/v2\/folder?post=4349"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}