{"id":3874,"date":"2022-09-08T17:09:55","date_gmt":"2022-09-08T15:09:55","guid":{"rendered":"https:\/\/erlebnisland-mathematik.de\/?page_id=3874"},"modified":"2023-06-09T14:13:07","modified_gmt":"2023-06-09T12:13:07","slug":"advanced-text-dust-circles","status":"publish","type":"page","link":"https:\/\/erlebnisland-mathematik.de\/en\/advanced-text-dust-circles\/","title":{"rendered":"Advanced text dust circles"},"content":{"rendered":"

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

Dust circles<\/h1>\n

The subject of the exhibit “Dust Circles<\/a>” is an interesting mathematical theorem about plane movements. First of all, you can look at the exhibit: It consists of two transparent plastic plates, each of which has black dust embedded in a congruent manner. The upper of the two is clamped in a wooden frame that can be moved against the lower fixed panel. What can you observe now?<\/p>\n

It seems as if circles<\/em> are involuntarily formed by this displacement — as the name of the exhibit already reveals.[\/vc_column_text][vc_single_image image=”1030″ img_size=”large” alignment=”center”]Figure 1: Actuation of the exhibit[\/vc_single_image][vc_column_text]<\/p>\n

And now the mathematics:<\/h3>\n

[\/vc_column_text][vc_column_text]But what is the reason for this phenomenon? For this purpose, we want to introduce a few mathematical terms that enable a precise description of what is observed. To do this, we first introduce the so-called Euclidean metric<\/em><\/a> on the plane \"\mathbb R^2\": Let \"\mathbf p=(p_1,p_2)\" and \"\mathbf q=(q_1,q_2)\" be two points in the plane. Then their Euclidean distance is defined as <\/p>\n

  <\/span>   <\/span>\"\[d(\mathbf p,\mathbf q)\coloneqq\sqrt{(p_1-q_1)^2+(p_2-q_2)^2}.\]\"<\/p>\n

A mapping<\/em><\/a> \"f\colon\mathbb R^2\to\mathbb R^2\" of the plane in itself that preserves the Euclidean distance, i.e. it is \"d(f(\mathbf p),f(\mathbf q))=d(\mathbf p,\mathbf q)\" for all \"\mathbf p,\mathbf q\in\mathbb R^2\", is called isometry<\/a><\/em>. Why do we look at such illustrations? Well, the reasoning lies in the fact that moving the upper plastic disk against the lower one corresponds exactly to such a distance-preserving self-imaging of the plane, because in doing so the distance between two points remains the same, regardless of whether we move the disk.<\/p>\n

The question that the exhibit indirectly asks the visitor is this: What distance-preserving self-images of the Euclidean plane exist at all?<\/em><\/p>\n

We want to fully clarify this question in the course of this short in-depth text. To do this, we first observe that any translation<\/a><\/em> around a given vector \"\mathbf v=(v_1,v_2)\in\mathbb R^2\" is distance-preserving, because it holds <\/p>\n

  <\/span>   <\/span>\"\begin{align*} d(\mathbf p+\mathbf v,\mathbf q+\mathbf v)&=\sqrt{(p_1+v_1-(q_1+v_1))^2+(p_2+v_2-(q_2+v_2))^2} \\ &=\sqrt{(p_1-q_1)^2+(p_2-q_2)^2}=d(\mathbf p,\mathbf q). \end{align*}\"<\/p>\n

But the translations are now in bijection<\/em><\/a> with the vectors \"\mathbf v\in\mathbb R^2\" themselves. So if \"f\colon\mathbb R^2\to\mathbb R^2\" is a distance-preserving mapping, we can form the mapping \"f'=f-f(\mathbf 0)\", which still preserves the Euclidean distance but now fixes the zero point \"\mathbf 0\". Therefore, we may assume that \"f(\mathbf 0)=\mathbf 0\". Now let us consider the images of the two standard unit vectors<\/em><\/a> \"\mathbf e_1\" and \"\mathbf e_2\" under \"f\". The right triangle \"\triangle(0,\textbf{e}_1,\textbf{e}_2)\" is determined by the lengths of its sides (namely \"1\", \"1\" and \"\sqrt{2}\") up to congruence<\/em><\/a>. However, these are preserved under \"f\", so that \"f(\mathbf e_1)\" and \"f(\mathbf e_2)\" must again be unit vectors perpendicular to each other. Consider further that each vector \"\mathbf v=(v_1,v_2)\in\mathbb R^2\" is uniquely determined by its distances to the three points \"\mathbf 0\", \"\mathbf e_1\", \"\mathbf e_2\". Namely, these are exactly <\/p>\n

  <\/span>   <\/span>\"\[a=\sqrt{v_1^2+v_2^2}, b=\sqrt{(v_1-1)^2+v_2^2}, c=\sqrt{v_1^2+(v_2-1)^2},\]\"<\/p>\n

so that \"a^2-b^2=2v_1-1\" and \"a^2-c^2=2v_2-1\" uniquely define the parameters \"v_1\" and \"v_2\". But with this \"f(\mathbf v)=v_1f(\mathbf e_1)+v_2f(\mathbf e_2)\" must be, because this vector has the same distances to \"\mathbf 0=f(\mathbf 0)\", \"f(\mathbf e_1)\", \"f(\mathbf e_2)\" as \"\mathbf v\" to \"\mathbf 0,\mathbf e_1,\mathbf e_2\". Thus, the mapping \"f\" must now even be linear<\/em><\/a> (because this corresponds exactly to the observation just made). But since \"f\" also maps the orthonormal basis<\/em><\/a> \"\mathbf e_1,\mathbf e_2\" to such a basis again, \"f\" corresponds to an orthogonal \"2\times 2\" matrix. However, these matrices can now be described very simply: To do this, we assume that \"\mathbf e_1\" is mapped onto the unit vector \"\mathbf f_1=(a,b)\". Because \"a^2+b^2=1\", we find an angle \"\alpha\in[0,2\pi)\" such that \"a=\cos(\alpha)\" and \"b=\sin(\alpha)\" (since the equation describes exactly the unit circle<\/a><\/em>). The vector \"\mathbf e_2\" must now be mapped to a unit vector \"\mathbf f_2=(c,d)\" which is perpendicular to \"\mathbf f_1\". But it already follows that \"\mathbf f_2=\pm(-b,a)\", since there are only two such vectors. Thus \"\mathbf f_2=(c,d)=(\mp\sin(\alpha),\pm\cos(\alpha))\" holds. So for \"f\" we get the matrix <\/p>\n

  <\/span>   <\/span>\"\[M=\begin{pmatrix} \cos(\alpha) & -\varepsilon\sin(\alpha)\\ \sin(\alpha) & \varepsilon\cos(\alpha)\end{pmatrix}.\]\"<\/p>\n

Here \"\varepsilon\in\{\pm1\}\" can be chosen arbitrarily. If \"\varepsilon=+1\", this simply corresponds to a rotation<\/em><\/a> by the angle \"\alpha\". This fact is the issue highlighted by the exhibit. If, on the other hand, \"\varepsilon=-1\", we obtain a reflection on the straight line \"g\" that intersects the \"x\" axis at an angle of \"\alpha/2\".<\/p>\n

Thus, we have fully understood the distance-preserving self-mappings of the Euclidean plane: Any such mapping \"f\colon\mathbb R^2\to\mathbb R^2\" is of the form <\/p>\n

  <\/span>   <\/span>\"\[f(\mathbf x)=D_{\alpha,\varepsilon}\mathbf x+\mathbf v\]\"<\/p>\n

for a rotation or reflection matrix \"D_{\alpha,\varepsilon}\" as above and a vector \"\mathbf v\" corresponding to the subtracted translation. However, if we start at the identical self-image (where \"\varepsilon=+1\" holds) and take it to a certain other distance-preserving image, the parameter \"\varepsilon\" cannot make a “jump<\/em>” (because otherwise the orientation would suddenly reverse and thus the plate would be flipped, which is not allowed in the exhibit). Thus we can restrict ourselves to the case \"\varepsilon=+1\" (if \"\varepsilon=-1\", we can show that then \"f\" always represents the reflection on a straight line). Now two cases can occur:<\/p>\n

Case 1<\/em>: \"f\" has a fixed point \"\mathbf x\". Then <\/p>\n

  <\/span>   <\/span>\"\[f(\mathbf x)=\mathbf x=D_{\alpha,1}\mathbf x+\mathbf v\]\"<\/p>\n

holds. From this follows \"(\mathrm{id}-D_{\alpha,1})\mathbf x=\mathbf v\". So \"f\" is then given by <\/p>\n

  <\/span>   <\/span>\"\[f(\mathbf y)= D_{\alpha,1}\mathbf y+\mathbf x-D_{\alpha,1}\mathbf x=D_{\alpha,1}(\mathbf x-\mathbf y)+\mathbf x,\]\"<\/p>\n

which is simply the rotation about the point \"\mathbf x\" by the angle \"\alpha\".<\/p>\n

Case 2<\/em>: \"f\" has no fixed point. Since the equation <\/p>\n

  <\/span>   <\/span>\"\[\mathbf x=D_{\alpha,1}\mathbf x+\mathbf v\]\"<\/p>\n

is always solvable for \"\alpha\neq 0\" (because \"\mathrm{id}-D_{\alpha,1}\" is then not singular<\/em><\/a>), consequently \"\alpha=0\" must hold. \"f\" is then simply a translation.<\/p>\n

The two cases 1 and 2 now completely clarify the phenomenon observed on the exhibit: Either one sees large dust circles (case 1) or at least dust straight lines (case 2).<\/p>\n

Finally, it should be noted that observed interesting fact for the group of all orientation-preserving (\"\varepsilon=+1\") self-mappings of the Euclidean plane lies group-theoretically<\/em><\/a> in the fact that it is a so-called Frobenius group<\/em><\/a>. This is expressed precisely in the fact that the elements that do not fix a point together with the identity form a subgroup (the translations).[\/vc_column_text][vc_column_text]<\/p>\n

Literature:<\/h2>\n

[1] https:\/\/en.wikipedia.org\/wiki\/3D_rotation_group<\/a><\/p>\n

[2] https:\/\/en.wikipedia.org\/wiki\/Isometry<\/a><\/p>\n

[3] https:\/\/en.wikipedia.org\/wiki\/Euclidean_space<\/a><\/p>\n

[4] https:\/\/en.wikipedia.org\/wiki\/Orthonormal_basis<\/a><\/p>\n

[5] https:\/\/en.wikipedia.org\/wiki\/Frobenius_group<\/a>[\/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] Dust circles The subject of the exhibit “Dust Circles” is an interesting mathematical theorem about plane movements. First of all, you can look at the exhibit: It consists of two transparent plastic plates, each of which has black dust embedded in a congruent manner. The upper of the two is clamped in … Advanced text dust circles<\/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-3874","page","type-page","status-publish","hentry"],"acf":[],"_links":{"self":[{"href":"https:\/\/erlebnisland-mathematik.de\/en\/wp-json\/wp\/v2\/pages\/3874","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=3874"}],"version-history":[{"count":20,"href":"https:\/\/erlebnisland-mathematik.de\/en\/wp-json\/wp\/v2\/pages\/3874\/revisions"}],"predecessor-version":[{"id":4658,"href":"https:\/\/erlebnisland-mathematik.de\/en\/wp-json\/wp\/v2\/pages\/3874\/revisions\/4658"}],"wp:attachment":[{"href":"https:\/\/erlebnisland-mathematik.de\/en\/wp-json\/wp\/v2\/media?parent=3874"}],"wp:term":[{"taxonomy":"folder","embeddable":true,"href":"https:\/\/erlebnisland-mathematik.de\/en\/wp-json\/wp\/v2\/folder?post=3874"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}