S S symmetry

Article A New or Math Porcelain Service

Valery Ochkov 1,2 , Massimiliano Nori 3 , Ekaterina Borovinskaya 4,5,* and Wladimir Reschetilowski 4,5

1 National Research University, Moscow Power Engineering Institute, 111250 Moscow, Russia; [email protected] 2 Joint Institute for High Temperatures of the Russian Academy of Science, 125412 Moscow, Russia 3 Department, Universtity or Institution, Viale delle Mimose 18, 64025 Pineto (TE), Italy; [email protected] 4 Technische Universität Dresden, 01062 Dresden, Germany 5 Saint-Petersburg State Institute of Technology (Technical University), 190013 St. Petersburg, Russia; [email protected] * Correspondence: [email protected]; Tel.: +49-351-463-33423

 Received: 9 December 2018; Accepted: 29 January 2019; Published: 4 February 2019 

Abstract: Egglipse was first explored by Maxwell, but Descartes discovered a way to modify the pins-and-string construction for to produce more egg-shaped curves. There are no examples of serious scientific and practical applications of Three-foci ellipses until now. This situation can be changed if porcelain and ellipses are combined. In the introduced concept of the egg-ellipse, unexplored points are observed. The new Three-foci ellipse with an equilateral triangle, a square, and a as “foci” are presented for this application and can be transformed by animation. The new elliptic-hyperbolic is presented. The other two similar curves, and parabola, can be also used to create new porcelain designs. Curves of the order of 3, 4, 5, etc. are interesting for porcelain decoration. An idea of combining of 3D printer and 2D colour printer in the form of 2.5D Printer for porcelain production and painting is introduced and listings functions in Mathcad are provided.

Keywords: ellipse; parabola; hyperbola; elliptic-hyperbolic oval; 2.5D Printer; augmented reality; programming; graphics; animation

1. Introduction The inventor of European white porcelain is Count Ehrenfried Walther von Tschirnhaus (1651–1708), who at the turn of the 17th and 18th centuries conducted experiments in Saxony on the creation of porcelain, and then organized its production in Meissen near Dresden. However, some historians believe that the real inventor of European white porcelain was not the aristocrat Tschirnhaus, but the monk-alchemist Johann Friedrich Böttger (1682–1719). Tschirnhaus kept him under arrest in the fortress. After the death of Tschirnhaus Böttger appropriated the laurels of the inventor of European porcelain and was locked up because he once tried to sell the secret of making porcelain to Prussia, however this attempt was suppressed. Böttger was an alchemist. These medieval “non-chemists” tried, in particular, to get the philosopher’s stone—a reagent necessary for the transformation of inexpensive metals into gold. The invention of porcelain somehow realized this dream—porcelain in those years, and even now is a very expensive commodity made from relatively cheap raw materials (kaolin, quartz, etc.), but bringing high profits, if it is manufactured with intelligence and talent. Prior to Tschirnhaus and Böttger in Europe there was only imported Chinese porcelain. Then porcelain was produced in Austria, France, England, Italy, Russia, the USA and in other countries. However Saxon porcelain is a special porcelain,

Symmetry 2019, 11, 184; doi:10.3390/sym11020184 www.mdpi.com/journal/symmetry Symmetry 2019, 11, 184 2 of 18 Symmetry 2019, 11, 184 2 of 18 a special porcelain, also because of the areola of the European “primogeniture”. Furthermore, European porcelain (hard porcelain) cannot be considered a replica of Chinese porcelain (soft porcelain).also because of the areola of the European “primogeniture”. Furthermore, European porcelain (hard porcelain)Tschirnhaus cannot beleft considered his mark a in replica mathematics of Chinese also. porcelain In Dresden (soft porcelain). there is a gymnasium with a mathematicalTschirnhaus inclination, left his which mark bears in mathematics the name of also. Count In Tschirnhaus. Dresden there He, isin aparticular, gymnasium explored with ellipsesa mathematical with three inclination, foci, which in which German bears literature the name are called: of Count The egg Tschirnhaus.-shaped ellipses He, of in Tschirnhaus particular, [1].explored This ellipse ellipses was with first three explored foci, whichby James in GermanMaxwell literaturein 1846. However, are called: Rene The Descartes egg-shaped discovered ellipses anof Tschirnhaus interesting [way1]. This to modify ellipse was the first pins explored-and-string by James construction Maxwell for in ellipses 1846. However, to produce Rene more Descartes egg- shapeddiscovered curves aninteresting [2,3]. Fresh way ideas to modifyfor porcelain the pins-and-string decoration can construction be generated for due ellipses to a to combination produce more of porcelainegg-shaped with curves ellipses [2,3].. Fresh ideas for porcelain decoration can be generated due to a combination of porcelainAt Meissen with ellipses. manufactory, all plates and saucers are only made round. Only large plates are made oval At(elliptical). Meissen Other manufactory, porcelain all factories plates and produce saucers plates are only and made saucers round. not Onlyonly largeround, plates but also are madeoval, squareoval (elliptical). or triangular Other with porcelain rounded factories corners produceand so on. plates Oval and(elongated) saucers plates not only are round, usually but served also oval,with fish.square However or triangular, Meissen with porcelain rounded is corners not the and porcelain so on. on Oval which (elongated) something plates is served are usually at the served table. withMeissen fish. porcelain However, is Meissenusually only porcelain admired is not or thebragged porcelain about on before which the something guests:”Anxious is served to at obliterate the table. theMeissen memory porcelain of that is emotion, usually onlyhe could admired think or of braggednothing aboutbetter than before china; the guests:”Anxious and moving with to her obliterate slowly fromthe memory cabinet ofto thatcabinet, emotion, he kept he couldtaking think up bits of nothingof Dresden better and than Lowestoft china; and and moving Chelsea, with turning her slowly them roundfrom cabinet and round to cabinet, with his he thin, kept veined taking uphand bitss, whose of Dresden skin, andfaintly Lowestoft freckled, and had Chelsea, such an turning aged look them?” [round4]. and round with his thin, veined hands, whose skin, faintly freckled, had such an aged look?” [4]. An ellipse is the of points on a plane, for each of which the sum of the distances from two other points, called foci, are constant [ 5–8].]. There is a simple simple way of of drawing drawing an ellipse: ellipse: T Twowo pins are fixedfixed into a sheet of paper, paper, a string is is attached attached to to them, them, along along which which a a pencil pencil slides slides,, drawing drawing an an ellipse. ellipse. If two two pins pins stick stick to to one one point point (take (take only only one one pin), pin), then then a circle a circle will be will drawn be drawn—the—the locus of locus points of pointsequidistant equidistant from the from center the of center the circ ofle. the If circle. one take If ones three takes pins three and pinsstick ands them sticks into them three into different three pointsdifferent (foci), points one (foci), can one draw can drawan ellipse an ellipse with with three three foci foci—the—the egg egg-shaped-shaped ellipse ellipse investigated investigated by Tschirnhaus (Figures(Figures1 1 and and2 ).2).

FigureFigure 1. 1.Drawing Drawing aa three-focithree-foci ellipse, ellipse, L L11+ + 2L2L22 + L 3 == const const..

The rope is fixedfixed with a pin to point F1,, then then the the pencil pencil lead lead is is tossed tossed over over through through the the pin pin stuck at point F2 and fixed fixed to point FF33. Moving Moving the pencil andand thusthus changingchanging thethe lengthslengths LL11,L, L22, and L33, but keeping the sum of L1 + 2L 2L22 ++ L L3 3constant,constant, it it is is possible possible to to draw draw the the closed closed curves curves shown shown in in Fig Figureure 2.. Authors will not use in the furtherfurther text thethe 22 beforebefore LL22. This trick was obtained because the authors used a model with a string in FigureFigure1 1 with with aa doubledouble stringstring betweenbetween thethe secondsecond focusfocus andand thethe pencil.pencil. TwoTwo-foci-foci ellipses have important scientificscientific applications—manyapplications—many natural and artificialartificial celestial objects (planets andand theirtheir satellites) satellites) move move in in elliptical elliptical orbits orbits or or in in orbits orbits close close to circularto circular ones. ones. Three-foci Three- fociellipses, ellipses as mentioned, as mentioned in the introduction,in the introducti do noton, havedo not any have serious any scientific serious and scientific practical and application. practical Thisapplication. situation This can situation be fixed, can if you be fixed think, if that you the think mathematician that the mathematician Tschirnhaus Tschirnhaus attended the attended creation the of creationEuropean of porcelain. European porcelain. Figure2 2 shows shows a a sketch sketch of of a porcelaina porcelain plate plate with with two two blue blue Three-foci Three-foci ellipses ellipses of Tschirnhaus. of Tschirnhaus. The plateThe plate itself itself is made is made in the in form the form of such of such an ellipse an ellipse (black (black contour contour of the of plate) the plate) [9]. [9].

Symmetry 2019, 11, 184 3 of 18 Symmetry 2019, 11, 184 3 of 18

Figure 2. Sketch of a porcelain plate with two blue t three-focihree-foci ellipsesellipses of Tschirnhaus.Tschirnhaus.

Three foci f11, f 2,, f f33, ,are are drawn drawn on on the the plate, plate, as as well well as as a a trace trace of of a stretched rope, the length of which remains constant when drawing the outer large ellipse of Tschirnhaus, on which one point is marked. ToTo drawdraw thethe same same internal internal ellipse, ellipse, the the rope rope was was shortened. shortened. The The distances distanc fromes from the focithe foci to one to ofone the of pointsthe points of the of outerthe outer ellipse ellipse are denotedare denoted by L by1,L L21,and L2 and L3. L These3. These parameters parameters can can be be estimated estimated in thisin this way: way: q 2 2 LL1 == (x − xf1) + (y − yyf1)) ==2424.59.59 cm cm,, q 2 2 LL2== ((x − xf2) + (y − yf2)) ==99.35.35 cm cm,, q 2 2 (1) LL3== ((xx − xf3) + (y − yyf3) ==1616.06.06 cm cm,, L1 + L2 + L3 = const = 50 cm. L + L + L = const = 50 cm. The sum of these distances remains constant. For the outer ellipse, this distance is 50 cm, for the The sum of these distances remains constant. For the outer ellipse, this distance is 50 cm, for the inner ellipse it is 33 cm, and for the edge of the plate it is 52.5 cm. inner ellipse it is 33 cm, and for the edge of the plate it is 52.5 cm. 2. Materials and Methods 2. Materials and Methods The first idea of the article was to show a scanning method, which uses one flat rectangular The first idea of the article was to show a scanning method, which uses one flat rectangular area area and allows generation of new unusual and interesting curves with very simple properties. and allows generation of new unusual and interesting curves with very simple properties. The The traditional mathematical methods for analysis, based on the symbolic solution of and traditional mathematical methods for analysis, based on the symbolic solution of equations and systems, cannot always provide the solution method to generate curves with given properties. equation systems, cannot always provide the solution method to generate curves with given The method proposed by the author is applicable for the generation of not only curves, but also properties. The method proposed by the author is applicable for the generation of not only curves, geometric figures, i.e., if not equations, but inequalities are considered. This method is also applicable but also geometric figures, i.e., if not equations, but inequalities are considered. This method is also for the generation of geometrical bodies with given properties. applicable for the generation of geometrical bodies with given properties. The second idea of the article was to provide the scanning method for the generation of geometric The second idea of the article was to provide the scanning method for the generation of objects, which is optimally correlated with the currently widely used 2D and 3D printing technology. geometric objects, which is optimally correlated with the currently widely used 2D and 3D printing The authors introduced a concept of a 2.5D printer. technology. The authors introduced a concept of a 2.5D printer. The inventors and researchers of new curves used as a rule, the analytical method for solving The inventors and researchers of new curves used as a rule, the analytical method for solving the problems. This significantly limits the search capabilities and is very time consuming. Numerical the problems. This significantly limits the search capabilities and is very time consuming. Numerical methods, one of which is proposed by the authors, opens new possibilities in this search, but does not methods, one of which is proposed by the authors, opens new possibilities in this search, but does deny analytical methods. not deny analytical methods. In Figure2, three foci are marked, not by points, but by small . And this is In Figure 2, three foci are marked, not by points, but by small circles. And this is understandable—the point in its mathematical representation will not be visible on the plate. In this understandable—the point in its mathematical representation will not be visible on the plate. In this connection, there was an idea which broadens the notion of an ellipse. What if, in constructing ellipses, connection, there was an idea which broadens the notion of an ellipse. What if, in constructing one does not rely on point-foci, but on circles and draws an ellipse with such a new property. Two, ellipses, one does not rely on point-foci, but on circles and draws an ellipse with such a new property. three, four or more circles are drawn on the plane, and then a curve is drawn, the sum of the distances Two, three, four or more circles are drawn on the plane, and then a curve is drawn, the sum of the distances from the curve points to the circles-foci (guiding circles) are constant. This is a directrix

Symmetry 2019, 11, 184 4 of 18

Symmetry 2019, 11, 184 4 of 18 from the curve points to the circles-foci (guiding circles) are constant. This is a directrix curled into a circle.curled Theinto distancea circle. fromThe distance a point tofrom a circle a point is easy to a tocircle determine—the is easy to determine length of— athe straight length line of a between straight theline point between itself the and point the itself point and on thethe circlepoint lyingon the on circle the linelying joining on the the line point joining to thethe centrepoint to of the the centre circle. Theof the centre circle. of The the centre circle, of the the point circle, on the its circumferencepoint on its circumference and the point and on the our point new ellipseon our mustnew ellipse lie on onemust straight lie on one line. straight Instead line. of circle-foci, Instead of other circle closed-foci, other curves: closed Ellipses, curves: triangles, Ellipses, squares, triangles, rhombuses, squares, rectanglesrhombuses, (polygons), rectangles (polygons), straight lines, straight etc. can lines, be used.etc. can The be distanceused. The from distance a point from to a these point curves to these is alsocurves easy is also to determine. easy to determine. For example, For example, a point on a thepoint contour on the of contour a triangle of a or triangle of a square or of thata square is closest that tois closest a point, to that a poin is nott, that on this is not contour, on this can contour, be either can on be the either vertex on ofthe the vertex triangle of the or oftriangle the square or of (see the thesquare triangle (see the in Figuretriangle3) in or Figure on a segment 3) or on a perpendicular segment perpendicular to the side to of the the side triangle of the ortriangle of the or square of the (Figuresquare (Figure3). It is easy3). It is to easy create to appropriatecreate appropriate procedures procedures or functions or functions and use and them use them when when building building our newour newellipses. ellipses. A Mathcad A Mathcad document document with functions with functions returning returning the coordinates the coordinates of a point on of segments a point on of lines,segments circles, of lines,squares circles, and equilateralsquares and triangles equilateral closest triangles to the given closest point to the is stored given as point Supplementary is stored as Materialssupplementary (File_S1.xmcd). material (File_S1.xmcd The codes are). shownThe codes below. are shown below.

FigureFigure 3. 3.New New Three-foci Three-foci ellipse ellipse L Lc c+ + LLss + L tt == 1.80 1.80 m. m.

3. Results and Discussion

3.1. New Type of Ellipses 3.1. New Type of Ellipses Using as a of the ellipse a circle instead of a point does not change anything in the form of Using as a focus of the ellipse a circle instead of a point does not change anything in the form of the ellipse if the ellipse does not intersect the circle-focus. This occurs when the value of the sum of L the ellipse if the ellipse does not intersect the circle-focus. This occurs when the value of the sum of1 + L2 (bifocal ellipse) or the sum of L1 + L2 + L3 (three focal ellipse) is sufficiently large. Therefore, the L1 + L2 (bifocal ellipse) or the sum of L1 + L2 + L3 (three focal ellipse) is sufficiently large. Therefore, authors conducted a study not only of a circle, but of an equilateral triangle and a square as the focus the authors conducted a study not only of a circle, but of an equilateral triangle and a square as the of the ellipse. The animation of this study is provided within the Supplementary Materials (Video S1). focus of the ellipse. The animation of this study is provided within the supplementary material (Video Figure3 shows a new previously unseen Three-foci ellipse, where an equilateral triangle, a square S1). and a circle act as “foci” (directing closed curves, directrixes). The square in Figure3 seems distorted Figure 3 shows a new previously unseen Three-foci ellipse, where an equilateral triangle, a due to optical illusion because of the intersection with the triangle and the circle. square and a circle act as “foci” (directing closed curves, directrixes). The square in Figure 3 seems By changing discretely, the sum of the distances from the points of the ellipse to fixed “foci”, distorted due to optical illusion because of the intersection with the triangle and the circle. families of ellipses can be obtained. Figures4 and5 show modified Tschirnhaus ellipses in blue. By changing discretely, the sum of the distances from the points of the ellipse to fixed “foci”, The blue closed curves have different sums of distances. The three “foci” of these ellipses are the three families of ellipses can be obtained. Figures 4 and 5 show modified Tschirnhaus ellipses in blue. The red circles. blue closed curves have different sums of distances. The three “foci” of these ellipses are the three red circles.

Symmetry 2019, 11, 184 5 of 18 Symmetry 2019,, 11,, 184184 5 of 18

FigureFigure 4. 4. SketchSketch of of a a porcelain plate with modifiedmodified ellipses of Tschirnhaus (different(different blue blue ovals ovals

correspondcorrespond toto differentdifferent valuesvalues ofof thethe sumsum L Lcircle1circle1circle1 ++ L Lcircle2circle2circle2 + L+circle3circle3 Lcircle3)).. ).

FigureFigure 5.5. SketchesSketches ofof thethe drawingdrawing ofof twotwo porcelainporcelain platesplates obtainedobtained with a circlecircle andand twotwo squaressquares asas

“foci”“foci” (different (different(different blue blueblue ovals ovalsovals correspond correspondcorrespondto toto different differentdifferent values valuesvalues of ofof the thethe sum sumsum L Lcirclecirclecircle + L Lsquate1squate1squate1 + L+ squate2squate2 L squate2).). ).

InIn FigureFigure4 4,4,, the thethe outer outerouter largest largestlargest ellipse ellipseellipse is isis the thethe traditional traditionaltraditional (normal) (no(normal)rmal) egg-shaped eggegg--shaped Tschirnhaus Tschirnhaus ellipse ellipse withwith threethree focusfocus points,points, locatedlocated atat thethe centerscenters ofof thethe threethree circles.circles. TheThe remainingremaining sixsix ellipsesellipses havehave deviationsdeviations fromfrom thethe traditionaltraditional ellipseellipse insideinside thethe circles-foci,circles--focifoci,, whichwhichwhich causecause aa specificspecific visualvisual effect.effect. ThisThis isis oneoneone ofofof thethethe goalsgoalsgoals ofofof finefinefine artartart ininin generalgeneralgeneral andandand ofof thethethe decorationdecorationdecoration ofofof porcelainporcelainporcelain in inin particular. particularparticular.. FigureFigure5 5 shows shows two two sketches sketches of of a a TschirnhausTschirnhaus mathematical mathematical plate plate obtained obtained with with a a circle circle and and two two squaressquares inin thethe rolerole ofof foci.foci.i. On On thethe rightright right-hand--hand figure figure,figure,, a kind of horse or zebra zebra with longitudinal, longitudinal, instead ofof transverse,transversetransverse,, strips cancan bebe seen.seen.. The The foci foci in in Figures 4 44 and andand5 55can cancan be bebe likened likenedlikened to toto certain certaincertain lenses, lenseslenses which,, which distortdistort thethe images.imagesimages.. TheThe tasktask ofof determiningdetermining thethe distancedistance fromfrom aa givengiven pointpoint toto anan ellipseellipse isis itself quitequite difficultdifficulticult eveneven forfor aaa simplesimplesimple two-focitwotwo--focifoci ellipse.ellipse.ellipse. However, the the methodsmethods for for constructingconstructing curves curves with with given given properties, properties, describeddescribed below,below allow,, allow allow us us to solvetoto solve solve this thisproblem this problem problem in a simple in in a a simple simple way, which way, way, consists which which ofconsists finding of the fifinding minimum the elementminimum of element a vector. of a vector. CombiningCombining figuresfigures (closed(closedcurves), curves), which which are are used used as as “foci”, “foci” and,, andand their theirtheir mutual mutualmutual arrangement, arrarrangement, it isit possibleisis possiblepossible to to manufactureto manufacturemanufacture and and to to colorto color plates plates with with a non-repetitive a non--repetitiverepetitive mathematical mathematicalmathematical pattern. pattern.pattern. IfIf aa circlecircle is is used used as as a focusa focusfocus instead insteadinstead of aofof point, aa pointpoint a classic,, a classic ellipse ellipse (see Figure(see(see FigureFigure6a) or 6a)6a) a new oror aa closed newnew closedclosed curve (elliptic-hyperboliccurve (elliptic(elliptic--hyperbolic oval), which oval)),, consists whichwhich consist of a parts of of a an part ellipse of an and ellipse a part and of one a part hyperbola of one branch hyperbola (see Figurebranch6 b),(see can Figure be obtained. 6b),, can Suchbe obtained an elliptic-hyperbolic.. SuchSuch anan ellipticelliptic oval--hyperbolic is observed oval ifthe isis observed sum of distances ifif thethe sumsum from ofof itsdistances points tofrom the itsits focus points point to the and focus to the point focus and circle to the is sufficiently focus circle small is sufficiently and the elliptic-hyperbolic small and the elliptic oval-- crosseshyperbolic the circle.oval crosses thethe circle.circle.

Symmetry 2019, 11, 184 6 of 18

Symmetry 20192019,, 1111,, 184184 66 of of 18 Symmetry 2019, 11, 184 6 of 18

Figure 6. An Ellipse (a - the blue oval) and an elliptic-hyperbolic oval (b - the blue oval); the brown

Figurecircle is 6 the. An second Ellipse foc (aus - ,the the blue first ovalfocus) and is a anpoint elliptic. -hyperbolic oval (b - the blue oval); the brown FigureFigure 6. 6An. An Ellipse Ellipse (a (a- the- the blue blue oval) oval and) and an an elliptic-hyperbolic elliptic-hyperbolic oval oval ( b(b- the- the blue blue oval); oval) the; the brown brown circle is the second focus, the first focus is a point. 3.2. Animationcirclecircle is i thes the and second second Augmented focus, focus the, the Reality first first focus focus is is a a point. point. 3.2.3.2. Animation and Augmented Reality 3.2. FigureAnimation 7 shows and Augmented a Tschirnhaus Reality plate with the three-foci ellipses formed by a red circle, a blue squareFigure, and7 7a shows greenshows equilateral aa TschirnhausTschirnhaus triangle plateplate (see withwith also thethe Figure three-focithree 4).-foci However, ellipsesellipses formedonformed the plate byby aa only redred circle,onecircle, modified aa blueblue Figure 7 shows a Tschirnhaus plate with the three-foci ellipses formed by a red circle, a blue square,squarethree-foci, and ellipse a green can equilateral be put and triangle shown (seein the also animation. Figure4 4).). However,However,on on thethe plateplate onlyonly oneone modifiedmodified square, and a green equilateral triangle (see also Figure 4). However, on the plate only one modified three-focithree-foci ellipseellipse cancan bebe putput andand shownshown inin thethe animation.animation. three-foci ellipse can be put and shown in the animation.

Figure 7. Another sketch of a Tschirnhaus plate; the green triangle is a first focus, the dark blue

FiguresquareFigure 7. is Another7 .the Another second sketch sketch foc ofus, aof the Tschirnhaus a Tschirnhausred circleplate; is a plate third the; greenthefoc usgreen; triangle blue triangle ovals is a have firstis a first focus,different foc theus, sum darkthe darkL bluec + L blue squares + L t. Figure 7. Another sketch of a Tschirnhaus plate; the green triangle is a first focus, the dark blue issquare the second is the focus, second the foc redus, circle the red is a circle third is focus; a third blue foc ovalsus; blue have ovals different have sumdifferent Lc + sum Ls + Ltc .+ Ls + Lt. At squarethe moment, is the second a new foc trendus, the of red Information circle is a third Technology focus; blue calledovals have augmented different rsumeality Lc +is L sbooming: + Lt. MuseumAt the visitors moment, look a new at the trend exhibit of InformationI nformationthrough their TechnologyTechnology tablets or called smartphones augmentedaugmented and r realityeality receive is booming: not only At the moment, a new trend of Information Technology called augmented reality is booming: MuseumMadditionaluseum visitors audio and look video at the information, exhibit through but also their see the tablets object or in smartphones a new perspective and receiveor in a new not way. only Museum visitors look at the exhibit through their tablets or smartphones and receive not only additionalA person audio who and sees video a mathematical information, butTschirnhaus also see the plate object on inthe a new wall, perspective perspective on a table or in in a aa new newmuseum way. way. additional audio and video information, but also see the object in a new perspective or in a new way. display,A personperson directs whowho his sees seestablet a mathematicala ormathematical smartphone Tschirnhaus Tschirnhaus to the plate plate plateand on the seeson wall, the on wall, onhis a screen tableon a or tablenot in aonly or museum in information a museum display, A person who sees a mathematical Tschirnhaus plate on the wall, on a table or in a museum directsdisplay,about the his directs mathematics tablet his or tablet smartphone and or the smartphone porcelain to the plate art to object,the and plate sees but onand also his seesanimation, screen on his not fivescreen only frames informationnot onlywhich information are about shown the display, directs his tablet or smartphone to the plate and sees on his screen not only information mathematicsaboutin Figure the 8mathematics. and the porcelain and the art porcelain object, but art alsoobject, animation, but also five animation, frames whichfive frames are shown which in are Figure shown8. about the mathematics and the porcelain art object, but also animation, five frames which are shown in Figure 8. in Figure 8.

Figure 8.8. Frames ofof thethe drawingdrawing animationanimation on on a a Tschirnhaus Tschirnhaus plate plate at at (L (Lc +c + L Ls s+ + LLtt = S) S):: ( a) S = 1 1.920.920 m;

((b)) SS == 1.3601.360 m; (cc)) SS == 0.9100.910 m;m; ((d)) SS == 0.620 m; (e) SS == 0.330 m (the green triangle is a firstfirst focus,focus, thethe Figure 8. Frames of the drawing animation on a Tschirnhaus plate at (Lc + Ls + Lt = S): (a) S = 1.920 m; Figure 8. Frames of the drawing animation on a Tschirnhaus plate at (Lc + Ls + Lt = S): (a) S = 1.920 m; blue(b) S square= 1.360 is ism; thethe (c ) secondsecond S = 0.910 focusfoc mus; andand(d) S thethe = 0.620 redred circlecircle m; (e isis) S aa third=third 0.330 focus).foc mus (the). green triangle is a first focus, the (b) S = 1.360 m; (c) S = 0.910 m; (d) S = 0.620 m; (e) S = 0.330 m (the green triangle is a first focus, the blue square is the second focus and the red circle is a third focus). blue square is the second focus and the red circle is a third focus).

Symmetry 2019, 11, 184 7 of 18 Symmetry 2019, 11, 184 7 of 18 Figure 8 shows only one blue Tschirnhaus ellipse, which changes its length and forms intricate closedFigure curves,8 shows tearing only into one parts blue at Tschirnhaus some point ellipse, during which the animation changes its. This length animation and forms is provided intricate closedwithin curves,the supplementary tearing into material parts at ( someVideo point S1). In during the shrinking the animation. ellipse, Thisa secret animation meaning is associated provided withinwith the the practical, Supplementary rather than Materials the decorative (Video S1). function In the of shrinking the plate ellipse, can be a seen secret. The meaning vanishing associated ellipse withis the the food practical, eaten from rather the than plate. the decorative function of the plate can be seen. The vanishing ellipse is the foodIn this eaten augmented from the plate.reality, one can also make interactivity. The owner of the tablet can change the parametersIn this augmented of the expanded reality, one Tschirnhaus can also make ellips interactivity.e (the number The and owner of of the the tablet “foci”, can their change relati theve parametersposition, the of length the expanded of the “string” Tschirnhaus, etc.) and ellipse create (the new number sketche ands shapefor the of plate. the “foci”, If such their “creativity” relative position,takes place the inlength a museum of the with “string”, porcelain etc.) manufacture, and createnew then sketchesthis sketch for can the be plate. immediately If such “creativity”transferred takesto a white place plate in a museumand to get with its own porcelain author's manufacture, porcelain thenartwork. this sketch can be immediately transferred to a white plate and to get its own author’s porcelain artwork. 3.3. Tschirnhaus Watch 3.3. Tschirnhaus Watch Porcelain plates are often hung on the walls to decorate the interior of the rooms. Sometimes suchPorcelain plates are plates made are in the often form hung of ona wall the clock: walls toM decorateake a hole the in interiorthe centre of theof the rooms. plate Sometimes through wh suchich platesthe axes are for made hands in the pass. form These of awall clock clock:-plates Make can aalso hole be in mad the centree in the of thestyle plate of the through Tschirnhaus which the ellipse axes for(Figure hands 9) pass.. These clock-plates can also be made in the style of the Tschirnhaus ellipse (Figure9).

FigureFigure 9.9. WatchWatch withwith an an elliptical elliptical three-focal three-focal dial dial (blue (blue lines lines are are the the hours hour hands hand and and black black lines lines are are for thefor minutesthe minute hand).s hand). Figure9 takes us back to Figure1, where a pencil moves along the closed egg-shaped curve, Figure 9 takes us back to Figure 1, where a pencil moves along the closed egg-shaped curve, drawing the Tschirnhaus ellipse. Figure9 shows a moving triple of straight lines: The hour hand (blue drawing the Tschirnhaus ellipse. Figure 9 shows a moving triple of straight lines: The hour hand lines: Twelve and a half hours) and the minute hand (black lines: Thirty minutes). (blue lines: Twelve and a half hours) and the minute hand (black lines: Thirty minutes). 3.4. Parabola and Hyperbola 3.4. Parabola and Hyperbola The ellipse is one of the three curves of the second order. The other two similar curves are the The ellipse is one of the three curves of the second order. The other two similar curves are the hyperbola and the parabola, which can also be used for porcelain applications. hyperbola and the parabola, which can also be used for porcelain applications. A porcelain vase can be made in the form of a parabola (paraboloid), in which the stand is A porcelain vase can be made in the form of a parabola (paraboloid), in which the stand is the the directrix, and the top handle is the focus (Figure 10). The parabola is the geometric locus of directrix, and the top handle is the focus (Figure 10). The parabola is the geometric locus of points points equidistant from the focus of the parabola and from a straight line called the parabola directrix equidistant from the focus of the parabola and from a straight line called the parabola directrix (Figure 11). (Figure 11).

Symmetry 2019, 11, 184 8 of 18 SymmetrySymmetry2019 2019,,11 11,, 184184 88 of 1818 Symmetry 2019, 11, 184 8 of 18

Figure 10. Profile of the vase in the form of a parabola (red curve), a directrix (bottom black straight Figure 10. Profile of the vase in the form of a parabola (red curve), a directrix (bottom black straight line), a focus (top ball), and a vertical bar connecting all parts. Figureline), a 10.10focus. ProfileProfile (top ofball) the, and vase a invertical the form bar ofconnecting a parabola all (red parts curve),. a directrix (bottom black straight line), a focus (top ball) ball),, and a vertical bar connecting all parts.parts.

Figure 11. Parabola with a focus and a directrix (Dir) L1 = L2 = 2.235 (the red line is the parabola, the FigureFigure 11.11. ParabolaParabola withwith aa focusfocus andanda a directrix directrix (Dir) (Dir) L L1 1= = LL22 = 2.235 (the red line is thethe parabola,parabola, thethe black point is the focus and the dashed line is the directory). Figureblackblack pointpoint 11. P isarabolais thethe focusfocus with andand a focus thethe dasheddashed and a linedirectrixline isis thethe (Dir) directory).director L1 = yL)2. = 2.235 (the red line is the parabola, the black point is the focus and the dashed line is the directory). However,However,, asas aa focusfocus ofof thethe parabola,parabola, oneone couldcould alsoalso take a circle circle insteadinstead of of a a point point ( (Figure(Figure 12)),),, wherewhereHowever four four frames frames, as a of focus of the the animation of animation the parabola, of the of movement the one movement could of also the oftake directrix thethe a directrix directrixcircle (blue insteadline) (blue (blue of of line) line)the a point parabola of of the the (Figure parabola parabola through 12), whereitsthroughthrough focus four circleitsits focusfocus frames (black circlecircle of curve) the (black(black animation are curve)curve) shown. are of The theshown new movement.. “parabola”TheThe newnew of “parabola” “parabola”the (red directrix curve) (red(red is (blue broken curve)curve) line) is intois broken ofbroken two the branches, parabola intointo twotwo throughandbranches, then its remerges and focus then circle intoremerges (black one line. into curve) one are line. shown . The new “parabola” (red curve) is broken into two branches, and then remerges into one line.

Figure 12. Frames of the animation of the parabola with a circular focus (the red lines are the FigureFigure 12. 12.Frames Frames of of the the animation animation of the of parabola the parabola with a with circular a circular focus (the focus red lines(the redare the lines parabolas, are the parabolas, the black circles are the foci and the blue lines are the directories). Figuretheparabola black 12s circles., theFrames black are ofthe circles the foci animationare and the the foc blue i of and lines the the parabola are blue the line directories). withs are athe circular director focusies). (the red lines are the parabolaFocus ins, the black German circles language are the foc (ini and the the native blue line languages are the ofdirector Tschirnhaus)ies). is der Brennpunkt— Focus in thethe German language (in the native language of Tschirnhaus) is der Brennpunkt— “burning point”. A parabola has this property—if a parallel beam of light falls on a parabolic mirror, “burningFocus point” in the.. AAGerman parabolaparabola language hashas thisthis property (inproperty the native—ifif aa parallel languageparallel beambeam of Tschirnhaus) ofof lightlight fallsfalls onon is deraa parabolicparabolic Brennpunkt mirror,mirror,— then the reflected beam converges into the focus. There is such a kind of fine arts—burning in the “burningthenthen thethe reflectedreflected point”. A beambeam parabola convergesconverges has this intointo property thethe focus.focus.—if ThereThere a parallel isis suchsuch beam a ki ofnd light of fine falls arts on— aburning parabolic in themirror, sun sun on a tree with the help of a magnifying glass or a parabolic mirror. Without fire, it is impossible thenon a thetree reflected with the beam help convergesof a magnifying into the glass focus. or Therea parabolic is such mirror. a kind Withoutof fine arts fire,—burning it is impossi in theble sun to to make porcelain—billets of porcelain products are burned in special furnaces. Tschirnhaus himself onmake a tree porcelain with the— billetshelp of of a porcelainmagnifying products glass or are a parabolicburned in mirror. special Without furnaces. fire, Tschirnhaus it is impossi himselfble to developed paraboloid mirror, an example of this is at the Dresden museum. makedeveloped porcelain paraboloid—billets mirror, of porcelain an example products of this are is burnedat the Dresden in special museum. furnaces. Tschirnhaus himself developed paraboloid mirror, an example of this is at the Dresden museum.

Symmetry 2019, 11, 184 9 of 18 Symmetry 2019, 11, 184 9 of 18

If a factor is introduced into the constitutive equation of the parabola and makes it different from If a a factor factor is is introduced introduced into into the the constitutive constitutive equation equation of the of theparabola parabola and m andakes makes it different it different from unity, then the parabola will either collapse into an ellipse or split into two branches of the hyperbola unity,from unity,then the then parabola the parabola will either will collapse either collapse into an ellipse into an or ellipse split into or splittwo branches into two of branches the hyperbola of the (Figure 13). This factor is called eccentricity [10]. A silhouette of a vase can be formed by two (hyperbolaFigure 13). (Figure This factor 13). This is called factor iseccentricity called eccentricity [10]. A silhouette [10]. A silhouette of a vase of a can vase be can formed be formed by two by (Figure 13). hyperbolastwo hyperbolas (Figure (Figure 13). 13).

Figure 13 13.. ProfileProfile of a vase for flowers flowers made in the form of a single single-sheeted-sheeted hyperboloid.

The hyperbola is the geometrical locus of points, such that the modulus of the difference of distances from each point to the two foci is constant. In the ellipse, the sum of these two quantities remains constant. However However,, Figure 13 shows not a focal, but a directional method for constructing a hyperbola, when when not not two two foci foci are are fixed fixed on the on plane, the plane, but only but one only focus one supplemented focus supplemented by the directrix. by the directrix.Figures 14 Figures and 15 show14 and two 15 branchesshow two of branches the hyperbola, of the hyperbola, when the focal when points the focal are replaced points are by circlesreplaced or byellipses. circles These or ellipses. new hyperboles These new also hyperboles represent also the represent profiles of the porcelain profiles vases.of porcelain vases.

Figure 14. Hyperbola with two circular foci |L1 −L2| = a, a = 1.7 m, r = 0.5 m, r1 = 0.7 m (the red Figure 14. Hyperbola with two circular foci |L1 −L2| = a, a = 1.7 m, r = 0.5 m, r1 = 0.7 m (the red circles Figurecircles are14. theHyperbola foci, the with blue two curves circular is two foci blanch |L1 − ofL2| the = a, hyperbola). a = 1.7 m, r = 0.5 m, r1 = 0.7 m (the red circles are the foci, the blue curves is two blanch of the hyperbola).

SymmetrySymmetry2019 2019, 11, ,11 184, 184 10 of10 18 of 18 Symmetry 2019, 11, 184 10 of 18

Symmetry 2019, 11, 184 10 of 18

FigureFigure 15. Hyperbola 15. Hyperbola with with two two square square foci foci |L |L11 −− LL2|2 |= =a, a,a = a 0.940 = 0.940 m (the m (the red redsquares squares are the are foci, the the foci, the Figure 15. Hyperbola with two square foci |L1 − L2| = a, a = 0.940 m (the red squares are the foci, the blueblueblue curves curves curves is twois is two two blanch blanch blanch of of of the the the hyperbola). hyperbola)hyperbola).. Figure 15. Hyperbola with two square foci |L1 − L2| = a, a = 0.940 m (the red squares are the foci, the If, inIfblue constructing, in curves constructing is two plane blanch plane curves of curves the hyperbola) with with the the. support support ofof twotwo foci,foci, instead of of using using the the sum sum (ellipse) (ellipse) or If, in constructing plane curves with the support of two foci, instead of using the sum (ellipse) the differenceor the difference (hyperbola), (hyperbola), the product the product is used, is used then, then a curve a curve is obtained, is obtained, that that is known is known as the as the Cassini or the Ifdifference, in constructing (hyperbola), plane curvesthe product with the is supportused, then of two a curve foci, insteadis obtained, of using that the is sum known (ellipse) as the (Figure 16). ovalCassini (Figureor the oval difference 16 ().Figure (hyperbola), 16). the product is used, then a curve is obtained, that is known as the Cassini oval (Figure 16).

Figure 16. Ovals of Cassini (red ovals). Figure 16. Cassini ovals can also have FiguremoreFigure than 1616..Ovals Ovals two foci, ofof Cassini Cassini and these (red (red ovals fociovals ovals). )may. ). not be points, but circles (see Figure 17), squares, triangles. CassiniCassiniCassini ovals ovals ovals can can can also also also have have have more more more than thanthan twotwo foci,foci, and and these these foci foci foci may may may not not not be be points, be points, points, but but circles but circles circles (see (see (see FigureFigureFigure 17 ),17) squares,17), squares,, squares, triangles. triangles. triangles.

Figure 17. Three frames of animation of the three-foci (L1·L2·L3 = a) blue Cassini oval with red circles

in the role of foci: (a) a = 0.073 m3;(b) a = 0.317 m3;(c) a = 1.003 m3. Symmetry 2019, 11, 184 11 of 18

Figure 17. Three frames of animation of the three-foci (L1·L2·L3 = a) blue Cassini oval with red circles in the role of foci: (a) a = 0.073 m3; (b) a = 0.317 m3; (c) a = 1.003 m3.

If one does not work with the sum (ellipse), the difference (hyperbola) or the product (Cassini’s Symmetryovals), but 2019 with, 11, 184 the ratio, one gets the so-called Apollonius circles. All these curves are also suitable11 of 18 for the design of the mathematical porcelain service. If, in the focal (two-foci) construction of an Symmetry 2019, 11, 184 11 of 18 ellipse,Figure hyperbola, 17. Three Cassini frames ovals,of animation or Apollonius of the three circles,-foci (L one1·L2· Lpoint3 = a)- bluefocus Cassini is replaced oval with by ared straight circles line, the newin the curves role of ( Figfociure: (a) 1a8 =) 0.can073 be m 3obtained; (b) a = 0..317 m3; (c) a = 1.003 m3. IfIn one Figure does 18 not, the work point with can the be sumreplaced (ellipse), by a thecircle, difference a square (hyperbola) or a triangle or and the productget new (Cassini’scurves for ovals),researchIf butone and withdo fores thenot drawings ratio,work onewith on getsporcelain. the thesum so-called (ellipse), In this Apolloniuscase, the differencea circle, circles. a square(hyp Allerbola) or these a triangle or curves the product can are alsointersect (Cassini’ suitable ands forovals),not the intersect designbut with with of thethe a mathematicalstraightratio, one line. gets A porcelain thestraight so-called service.line, inApollonius turn, If, in ca then focalbecircles. replaced (two-foci) All these by a construction curve,curves and are in ofalso particular, an suitable ellipse, hyperbola,forby a the parabola. design Cassini Figure of the ovals, 19mathematical shows or Apollonius a “new porcelain circles,ellipse” one service. with point-focus two If, foci, in th one ise replaced focal focus (two is bya- foci)point, a straight construction and line,the other the of new is an a curvesellipse,parabola. (Figure hyperbola, 18) can Cassini be obtained. ovals, or Apollonius circles, one point-focus is replaced by a straight line, the new curves (Figure 18) can be obtained. In Figure 18, the point can be replaced by a circle, a square or a triangle and get new curves for research and for drawings on porcelain. In this case, a circle, a square or a triangle can intersect and not intersect with a straight line. A straight line, in turn, can be replaced by a curve, and in particular, by a parabola. Figure 19 shows a “new ellipse” with two foci, one focus is a point, and the other is a parabola.

FigureFigure 18.18.Extended Extended ellipse,ellipse, hyperbola,hyperbola, CassiniCassini ovaloval andand ApolloniusApollonius circlecircle (red(red ovalsovals andand curves):curves): ((aa)) L + L = s, s = 0.5 m; (b) |L + L | = s, s = 0.15 m; (c)L ·L = s, s = 0.05 m2;(d)L /L = s or L /L = s L11 + L2 = s, s = 0.5 m; (b) |L11 + L2|2 = s, s = 0.15 m; (c) L1·L12 = 2s, s = 0.05 m2; (d) L1/L21 = s 2or L2/L1 =2 s s 1= 0.7 s(blue = 0.7 lines (blue are lines the are first the focus first and focus the and black the points black points are the are second the second focus) focus).. In Figure 18, the point can be replaced by a circle, a square or a triangle and get new curves for research and for drawings on porcelain. In this case, a circle, a square or a triangle can intersect and not intersect with a straight line. A straight line, in turn, can be replaced by a curve, and in particular, Figure 18. Extended ellipse, hyperbola, Cassini oval and Apollonius circle (red ovals and curves): (a) by a parabola. Figure 19 shows a “new ellipse” with two foci, one focus is a point, and the other is L1 + L2 = s, s = 0.5 m; (b) |L1 + L2| = s, s = 0.15 m; (c) L1·L2 = s, s = 0.05 m2; (d) L1/L2 = s or L2/L1 = s s = 0.7 a parabola.(blue lines are the first focus and the black points are the second focus).

Figure 19. An ellipse with two foci given by a point and a parabola.

Attempts by the authors to find information on the Internet about point and line on the plane did not lead to mathematical formulas, but to a book with the same title by Vasily Kandinsky with the subtitle "Contribution to the analysis of pictorial elements". This book was originally published in German with the title “Punkt und linie zu flache” when Kandinsky was teaching at the well-known Bauhaus (before inFigure Figure Weimar, 19.19. AnAn then ellipseellipse in Dessau). withwith twotwo foci True,foci givengiven in byKandinsby aa pointpointky andand and aa parabola.parabola. other Russian avant-garde

AttemptsAttempts byby thethe authorsauthors toto findfind informationinformation onon thethe InternetInternet aboutabout pointpoint andand lineline onon thethe planeplane diddid notnot leadlead toto mathematicalmathematical formulas,formulas, butbut toto aa bookbook withwith thethe samesame titletitle byby VasilyVasily KandinskyKandinsky withwith thethe subtitlesubtitle "Contribution "Contribution to to the the analysis analysis of of pictorial pictorial elements". elements". This Th bookis book was was originally originally published published in Germanin German with with the the title title “Punkt “Punkt und und linie linie zu zu flache” flache” when when Kandinsky Kandinsky was was teaching teaching at at the thewell-known well-known BauhausBauhaus (before (before in in Weimar, Weimar, then then in in Dessau). Dessau). True, True, in in Kandinsky Kandinsky and and other other Russian Russian avant-garde avant-garde

Symmetry 2019, 11, 184 12 of 18 SymmetrySymmetry 2019 2019, ,11 11, ,184 184 1212 ofof 1818 artists,artistsartists, , everythingeverything everything waswas was reducedreduced reduced onlyonly only toto to aestheticaesthetic aesthetic issues.issues. issues. WeWe We alsoalso also addedadded added mathematicsmathematics mathematics toto to thisthis this creativecreati creativeve process.process.process. However, However,However, the thethe phrase phrase phrase “point “point “point and and line and to line line the to plane”to the the plane” is plane” redundant isis redundant redundant in terms ofin in classical terms terms of of Euclidean classical classical :EuclideanEuclidean geometry: Thegeometry: points T andThehe poin thepoints straightts and and thethe line straight straight are always line line are inare onealways always definite inin one one plane. definite definite If a plane. similarplane. IfIf book aa similar similar was writtenbookbook was was by awrittenwritten sculptor, by by insteada a sculptor, sculptor, of a painter,insteadinstead of heof a woulda pain painter,ter, have he heentitled would would have ithave “Point entitled entitled and ait it straight “ “PoiPointnt lineand and ina a straight space”!straight andlineline onein in space space more”!”! remarka andnd one one from more more the remark remark point from offrom view the the ofpoint point mathematics. of of view view of of mathematics. Onmathematics. a picturesque On On a canvas,a pi picturesquecturesque we see canvas, canvas, not a straightwewe see see not line,not a a butstraight straight only aline, line, segment but but only only of a a straighta segment segment line, of of whicha a straight straight can alsoline,line, actwhich which as a can focalcan also also point. act act Figureas as a a focal focal 20 shows point. point. theFigureFigure metamorphosis 20 20 showsshows the the of a metamorphosis metamorphosis traditional ellipse, of of a when a traditional traditional one of its ellipse, ellipse, focus when turns when into one one an of of elongating its its focus focus turns segmentturns into into of an ana straightelongatingelongating line. segment segment of of a a straight straight line. line.

FigureFigureFigure 20.20 20. An.An An ellipseellipse ellipse withwith with aa a focusfocus focus changingchanging changing fromfrom from aa a pointpoint point toto to aa a lineline line segment.segment. segment.

FigureFigureFigure 2121 21 showsshows shows twotwo two sketchessketches sketches ofof of thethe the TschirnhausTschirnhaus Tschirnhaus plate.plate. plate. TheThe The usualusual usual three-focithree three-foci-foci ellipseellipse ellipse (the(the (the edgeedge edge ofof of thethethe plate—Figureplate plate——FigureFigure2 2) )2) is is is dissected dissected dissected by by by three three three straight straight straight lines, lines, lines, whose whose whose points points points of of of intersection intersection intersection serve serve serve as as as foci foci foci of of of thethethe ellipse.ellipse. ellipse. RedRed Red lineslines lines areare are formedformed formed byby by points,points, points, whosewhose whose sumsum sum (Figure(Figure (Figure 2121a) 21a)a) oror or productproduct product (Figure(Figure (Figure 2121b) 21b)b) ofof of thethe the distancesdistancesdistances toto to thethe the straightstraight straight lineslines lines remainsremains remains constant.constant. constant.

FigureFigureFigure 21.21 21.Sketches .Sketches Sketches of of of the the the Tschirnhaus Tschirnhaus Tschirnhaus plates plates plates with with with a: (a a): : Constant( a(a) )C Constantonstant sum sum orsum (b or )or product ( b(b) )product product of distances of of distances distances from afromfrom point a a topoint point three to to straightthree three straight straight lines. lines. lines. 3.5. Curves of Order Greater Than Two 3.5.3.5. Curves Curves of of Order Order Greater Greater Than Than Two Two The ellipse, the hyperbola, and the parabola are curves of the second order. It was interesting to TheThe ellipse, ellipse, the the hyperbola, hyperbola, and and the the parabola parabola are are curves curves of of the the second second order. order. It It was was interesting interesting to to know what curves of the order of 3, 4, 5, etc. are good for porcelain decoration. Figure 22 illustrates knowknow what what curves curves of of the the order order of of 3, 3, 4, 4, 5, 5, etc. etc. are are good good f orfor porcelain porcelain decoration decoration. .Figure Figure 22 22 illustrat illustrateses such a problem: Points are dashed randomly through a plane through which curves of different orders suchsuch a a problem: problem: PPointsoints are are dashed dashed randomly randomly through through a a plane plane through through which which curves curves of of different different ordersorders are are drawn. drawn. A A p parabolaarabola can can only only come come about about with with a a special special and and almost almost incredible incredible arrangement arrangement

Symmetry 2019, 11, 184 13 of 18

Symmetry 2019, 11, 184 13 of 18 are drawn. A parabola can only come about with a special and almost incredible arrangement of of fivefive points. points. The The probability probability of ofthe the ellipse ellipse falling falling out out (about (about 28%) 28%) can can be becons consideredidered a certain a certain new new mathematicalmathematical constant. constant. Through Through two twopoints points,, it is itpossible is possible to draw to draw a straight a straight lineline (a curve (a curve of the of thefirst first order),order), in five in five—an—an ellipse ellipse or two or two branches branches of a of hyperbola a hyperbola (the (the second second order), order), through through nine nine points points curvescurves of the of thethird third order, order, etc. etc. These These intricate intricate curves curves can can be be also also used used to to decorate decorate porcelain porcelain.. This This will will in in somesome way way resemble resemble the the so so-called-called blue blue onion onion pattern. pattern.

Figure 22. Curves of different orders: (a) 2; (b) 3; (c) 4; (d) 5; (e) 6; (f) 7; (h) 13; (i) 16; (j) 17. Figure 22. Curves of different orders: (a) 2; (b) 3; (c) 4; (d) 5; (e) 6; (f) 7; (h) 13; (i) 16; (j) 17. It is possible to see the modification of the pattern shown in the animation provided with the It is possible to see the modification of the pattern shown in the animation provided with the Supplementary Materials (Video S2). supplementary material (Video S2). 3.6. 2.5D Printer 3.6. 2.5D Printer Tschirnhaus plates are made by traditional technology from porcelain, and then painted by hand. However,Tschirnhaus they plates can be are printed made on by a traditional 3D printer, technology and then painted from porcelain, on an ordinary and then 2D painted colour printer by hand.(kitsch). However, These they two can printers be print caned be on combined a 3D printer, into one, and which then canpaint beed named on an a ordinary 2.5D printer. 2D colour printer (kitsch).The figures These and two closed printers curves can given be combined above were into not one, created which through can be an named analytic a 2.5D study print of functions,er. butThe by figures a simple and (“blunt”) closed curves scan of given a rectangular above were region not in created Cartesian through coordinates. an analytic Modern study printers of functions,similarly but form by atext simple or drawing—scan (“blunt”) scan of a sheeta rectangular of paper region and put in blackCartesian or coloured coordinates. dots Modern (rasters) in printers similarly form text or drawing—scan a sheet of paper and put black or coloured dots (rasters) in the right places. Figure 23 shows a part of Mathcad document (a virtual 2D printer) that depicts the egg-shaped Tschirnhaus ellipse.

Symmetry 2019, 11, 184 14 of 18

Symmetrythe right 2019 places., 11, 184 Figure 23 shows a part of Mathcad document (a virtual 2D printer) that depicts14 of the 18 Symmetryegg-shaped 2019, Tschirnhaus11, 184 ellipse. 14 of 18

Figure 23. Scanning a flat area for drawing a Tschirnhaus three-foci ellipse. FigureFigure 23 23.. ScanningScanning a a flat flat area area for drawing a Tschirnhaus three-focithree-foci ellipse. In the program in Figure 23, the for loop with parameter x, in which is nested a second for a loop In the program in Figure 23, the for loop with parameter x, in which is nested a second for a loop with Inparameter the program y, moves in Figure points 23 , inthe the for rectangular loop with parameter area from x,x 1in to which x2 and is from nested y1 ato second y2. Variables for a loop x1, with parameter y, moves points in the rectangular area from x1 to x2 and from y1 to y2. Variables xwith2, y1 parameterand y2, as well y, moves as other points quantities in the rectangularand functions area needed from forx1 to calculation, x2 and from are y 1set to iny2 .advance. Variables The x1, x1, x2, y1 and y2, as well as other quantities and functions needed for calculation, are set in advance. valuex2, y1 andof the y2, integeras well asvariable other quantitiesn that defines and functionsthe scan step needed can forbe calculation,changed, achieving are set in a advance.compromise The The value of the integer variable n that defines the scan step can be changed, achieving a compromise betweenvalue of accuracythe integer and variable duration n that of the defines computation. the scan stepIn the can double be changed, for loop, achieving the distances a compromise from the between accuracy and duration of the computation. In the double for loop, the distances from the currentbetween point accuracy of x andand yduration coordinates of the to comthe putation.first focus In (L the1), to double the second for loop, focus the (L distances2) and to thefrom third the current point of x and y coordinates to the first focus (L1), to the second focus (L2) and to the third focus focuscurrent (L 3point) are calculated.of x and y Insteadcoordinates of points to the as first foci, focus segments (L1), toof thecurves, second circles, focus squares, (L2) and tri toangles, the third etc. (L3) are calculated. Instead of points as foci, segments of curves, circles, squares, triangles, etc. can be canfocus be (L put.3) are Distances calculated. from Instead a point of pointsto these as curves foci, segments can be easily of curves, calculated circles, using squares, special triangles, functions etc. put. Distances from a point to these curves can be easily calculated using special functions created createdcan be put.by the Distances authors. from Available a point in to the these supplementary curves can be material easily calculated(File_S2.xmcd) using. Ifspecial the sum functions of the by the authors. Available in the Supplementary Materials (File_S2.xmcd). If the sum of the distances distancescreated by L the1, L 2authors. and L3 Availableturns out in to the be supplementary approximately material equal to ( File_S2 the given.xmcd) variable. If the a,sum then of thethe L1,L2 and L3 turns out to be approximately equal to the given variable a, then the coordinates of the coordinatesdistances L 1of, Lthe2 and current L3 turnspoint are out recorded to be approximately in the vectors X equal and Y, to whose the given length variable is increased a, then by one the current point are recorded in the vectors X and Y, whose length is increased by one (i ← i + 1, where i (icoordinates ← i + 1, where of the i iscurrent the index point of are the recorded vectors X in and the Y). vectors Then X the and vectors Y, whose X and length Y are is displayedincreased byon onethe is the index of the vectors X and Y). Then the vectors X and Y are displayed on the graph in the form of graph(i ← i +in 1, the where form i ofis thea desired index curveof the consistingvectors X andof points. Y). Then If these the vectors points areX and large Y areenough, displayed they mergeon the a desired curve consisting of points. If these points are large enough, they merge into a line. intograph a line.in the form of a desired curve consisting of points. If these points are large enough, they merge Figures 24–26 show the listings of the functions used to work with a new focus—with circle, into aFigures line. 24–26 show the listings of the functions used to work with a new focus—with circle, square and triangle. In addition, these functions are provided with the Supplementary Materials squareFigures and t2riangle.4–26 show In a theddition, listings these of the functions functions are used provided to work with with the a supplementarynew focus—with material circle, (File_S1.xmcd). (sFile_S1.xmcdquare and triangle.). In addition, these functions are provided with the supplementary material (File_S1.xmcd).

Figure 2 24.4. AA Mathcad Mathcad function function that that returns returns the the coordinates coordinates of a a point point on on the the circumference circumference of a circle withFigure radius 24. A r Mathcadand cent centre refunction at the pointthat returns (x(xc,, ycc)) closestthe closest coordinates to to a a given given of point pointa point with with on the thethe coordinates coordinatescircumference ( (x,x, yof y).) .a circle with radius r and centre at the point (xc, yc) closest to a given point with the coordinates (x, y).

Symmetry 2019, 11, 184 15 of 18 SymmetrySymmetry 20192019,, 1111,, 184184 1515 ofof 1818

FigureFigure 25.2255.. AA MathcadMathcad function functionfunction that thatthat returns returnsreturns the thethe coordinates coordinatescoordinates of ofof a aa point pointpoint on onon the thethe contour contourcontour of ofof a a squarea squaresquare with withwith a “radius”aa “radius”“radius” r (half rr (half(half the thethe length lengthlength of the ofof sidethethe sideside of the ofof square) thethe square)square) and centred andand centredcentred at the atat point thethe (xpointpoints, ys )((xx closestss,, yyss)) closestclosest to the to givento thethe pointgivengiven with pointpoint the withwith coordinates thethe coordinatescoordinates (x, y). ((xx,, yy))..

FigureFigure 26.2266.. AA MathcadMathcad functionfunction thatthat returnsreturns thethe coordinatescoordinates ofof aa pointpoint onon thethe contourcontour ofof anan equilateralequilateral triangletriangletriangle withwithwith aaa “radius”“radius”“radius” rrr (the(the(the distancedistance fromfromfrom thethethe centercentercenter ofofof thethethe triangletriangle tototo itsits vertex)vertex) andand centeredcecenteredntered atatat thethe pointpointpoint (x((xxttt,, yytt)) closest to the the given given point point with with coordinates coordinates (x,((xx,, y).yy))..

TheThe distancedistancedistance from fromfrom the thethe already alreadyalready constructed constructedconstructed ellipse ellipseellipse to toto a a givena givengiven point pointpoint is is easyis easyeasy to toto determine, determine,determine, bearing bearingbearing in mindinin mindmind that thatthat our ourour ellipse ellipseellipse is formed isis formedformed not notnot by byby an anan analytic analyticanalytic formula, formulaformula but,, butbut by byby two twotwo vectors vectorsvectors with withwith discrete discretediscrete values valuesvalues of coordinates.ofof coordinates. coordinates. Figure FigureFigure 27 shows 27 27 showsshows the body the the bodybody of the ofof double-cycle the the double double with--cyclecycle parameters with with parameters parameters x and y (seex x and and the y y headings (see (see the the ofheadingsheadings these cycles ofof thesethese in Figure cyclescycles 23 inin), FigureFigure which 2323 solves),), whichwhich this solvessolves problem: thisthis The problem:problem: coordinates TThehe coordinatescoordinates of the two closedofof thethe twotwo curves closedclosed are givencurvescurves by areare the givengiven vectors byby thethe (X1, vectorsvectors Y1) and ((X1X1 (X2,,, Y1Y1) Y2).) andand It ((X2X2 is, necessary, Y2Y2)).. ItIt isis necessarynecessary to find and toto tofindfind place andand in toto the placeplace vectors inin thethe X vectorsvectors and Y theXX andand coordinates YY thethe coordinatescoordinates of the new ofof curve thethe newnew whose curvecurve points whosewhose have pointspoints a certain havehave property aa certaincertain with propertyproperty respect withwith to the respectrespect two initialtoto thethe twotwo initialinitial (reference)(reference) curves.curves. InIn particular,particular, thethe sumsum ofof thethe distancedistancess fromfrom thethe pointspoints formingforming thethe newnew curvecurve toto thethe pointspoints ofof thethe referencereference curvescurves mustmust bebe constant.constant. ItIt turnsturns outout aa supersuper--ellipse,ellipse, shownshown inin

Symmetry 2019, 11, 184 16 of 18

Symmetry(reference)Symmetry 20192019 curves.,, 1111,, 184184 In particular, the sum of the distances from the points forming the new curve1616 ofof 18 to18 the points of the reference curves must be constant. It turns out a super-ellipse, shown in Figure 28. FigureWhenFigure solving2828.. WhenWhen this solvingsolving problem, thisthis problem, auxiliaryproblem, auxiliary vectorsauxiliary D1 vectorsvectors and D2 D1D1 are andand formed, D2D2 areare formed, storingformed, distances storingstoring distancesdistances from a given fromfrom apointa givengiven to pointpoint the points toto thethe of pointspoints the reference ofof thethe referencereference curves. In curves.curves. the vectors InIn thethe D1 vectorsvectors and D2, D1D1 it andand is easy D2,D2, toitit is findis easyeasy the toto minimum findfind thethe minimumvalues—correspondingminimum valuesvalues——correspondingcorresponding to the distance toto thethe from distancedistance the point fromfrom tothe thethe curve. pointpoint We toto the repeatthe curve.curve. this taskWeWe repeat analytically,repeat thisthis tasktask it is analytically,ratheranalytically, difficult itit isis to ratherrather solve, difficuldifficul while numericallytt toto solve,solve, whilewhile can benumericallynumerically without problems. cancan bebe withoutwithout problemsproblems..

FigureFigure 2727 27... TheThe procedureprocedure forfor formingforming thethe super super-ellipsesuper--ellipseellipse shown shown in in Figure Figure 28...

FigureFigureFigure 28. 2828Super-ellipse.. SuperSuper--ellipsellipse withe withwith “elliptical” “elliptical”“elliptical” foci, focifoci L,, 1 LL1+1 + +L LL222 = = const,const, LL33 + + LL L333 + + + LL5 L5 = 5= const=const const...

TheThe smallsmall threethree three-foci--focifoci ellipses ellipses shown shown in in Figure Figure 2828 areare in in turn turn focifoci foci,,, in in fact fact focifoci foci cancan can bebe be notnot not onlyonly only points,points, points, butbut new new ellipsesellipses with withwith focal focalfocal points-ellipses. pointspoints--ellipses.ellipses. There ThereThere may maymay be bebe a fractal aa fractalfractal [11 ][11][11] and andand fractals fractalsfractals themselves themselvesthemselves are very areare veryinterestingvery interestinginteresting objects, objects,objects, also for alsoalso the forfor decoration thethe decorationdecoration of porcelain ofof porcelainporcelain with elements withwith elementselements of mathematics. ofof mathematics.mathematics. However, HoweveHoweve thisr, isr, thisthethis topicisis ththee of topictopic a separate ofof aa separateseparate “mathematical-aesthetic” “mathematical“mathematical--aesthetic”aesthetic” conversation conversationconversation [12,13 ]. [12,[12,13].13]. n n TheThe supersuper super-ellipse--ellipseellipse oror or LameLame Lame ellipseellipse ellipse isis is anan an ellipseellipse ellipse describeddescribed described byby by thethe the equationequation equation (x/a)(x/a) (x/a)nn ++ (y/b)(y/b)+ (y/b)nn = = 1.1. ForFor= n 1.n =For= 2,2, nwewe = 2,havehave we haveanan ordinaryordinary an ordinary ellipse.ellipse. ellipse. FiguresFigures Figures 2299 and and29 and 3030 show30show show sketchessketches sketches forfor for thethe the designdesign design ofof of porcelainporcelain porcelain plates plates inin the the style style of of Lame Lame ellipses ellipses with with different different exponents exponentsexponents of of degree degreedegree n. n.n. In In the the right right parts parts of of the the drawing’s drawing’s informationinformation is isis given, given, which which willwill needneed toto bebe be placedplaced placed onon on thethe the reversereverse reverse sidessides sides ofof of thesethese these “mathematical”“mathematical” “mathematical” plates.plates. plates.

Symmetry 2019, 11, 184 17 of 18 Symmetry 2019, 11, 184 17 of 18 Symmetry 2019, 11, 184 17 of 18

FigureFigure 2 29.9.. LameLame round round plate. plate.

Figure 30. Lame Oval plate with golden section proportion. Figure 30.. Lame Oval plate with golden section proportion. 4. Conclusions 5. Conclusions 5. ConclusionsA scanning method, which is used in one flat rectangular area and allows the generation of new unusualA scanning and interesting method, curves which with is use veryd in simple one flat properties, rectangular was area introduced. and allows This the method generation is applicable of new unusualfor generation and interesting of not only curves curves, with but also very geometric simple figures,properties i.e.,,, ifwas was inequalities in introduced instead.. This This of method method equations is is applicableare considered. for generation of not only curves, but also geometric figures, i.e.,, ifif inequalities instead of equationsThe suggested are considered. method introduces a new stream into the design of porcelain dishes. The egg-ellipse can helpThe suggested to provide method new ideas introduce for porcelains a new decoration stream into and the to design find practical of porcelain as well dishes. as decorative The egg- eapplication.llipse can help The new to provide Three-foci new ellipse ideas (elliptic-hyperbolic for porcelain decoration oval) with and an to equilateral find practical triangle, as a well square as decorativeand a circle application. as “foci” is introduced. The new TheThree other-foci two ellipse second (elliptic order-hyperbolic curves, the hyperbola oval) with and an the equilateral parabola, tricanangle, be also a squareused to and create a a circle new porcelain as “foci” design.is introduced. Curves of The the other order two of 3, second 4, 5, etc. order can also curves, inspire the a hyperbolaporcelain designer. and the parabola An idea of,, can combination be also used of 3D to printercreate a and new 2D porcelain colour printer design. in Curves form of of 2.5D the printer order offor 3, porcelain 4, 5, etc. production can also inspire and decoration a porcelain is presented designer. and An theidea functions of combination codes in of Mathcad 3D printer are provided. and 2D colourThe super-ellipse printer in form or lame of 2.5 ellipseD printer can providefor porcelain ideas product for the designion and of decoration porcelain is plates presented in the and style the of functionslame ellipses codes with in Mathcad different exponentsare provided. of degree The super n. -ellipse or lame ellipse can provide ideas for the designThe of ideasporcelain proposed plates inin thethe style article of were lame presentedellipses with at thedifferent porcelain exponents factory of in degree Meissen. n. Factory representativesThe ideas proposed were interested in the in article the work were and presented plan to use at the mathematical porcelain drawingsfactory in onMeissen. porcelain Factory when representativesreceiving orders. were interested in the work and plan to use mathematical drawings on porcelain when receiving orders.

Symmetry 2019, 11, 184 18 of 18

Supplementary Materials: The following are available online at http://www.mdpi.com/2073-8994/11/2/184/s1, File_S1: Mathcad document with functions returning the coordinates of a point on segments of lines, circles, squares and equilateral triangles closest to the given point, File_S2: Calculation of distances from a point to the curves using special functions created by the authors, Video S1: Frames of drawing animation on a Tschirnhaus plate at (Lc + Ls + Lt = S), Video S2: Curves of different orders. Author Contributions: Conceptualization, V.O.; methodology, V.O.; software, M.N.; validation, V.O., M.N.; formal analysis, E.B., W.R.; investigation, V.O.; resources, E.B.; writing—original draft preparation, V.O., M.N., E.B., W.R.; writing—review and editing, M.N., E.B., W.R.; visualization, V.O, M.N., E.B., W.R.; supervision, W.R. Funding: This research received no external funding. Conflicts of Interest: The authors declare no conflict of interest.

References

1. Sahadevan, P.V. The theory of egglipse—A new curve with three focal points. Int. J. Math. Educ. Sci. Technol. 1987, 18, 29–39. [CrossRef] 2. Williamson, B. The properties of the treated geometrically. Hermathena 1876, 2, 509–518. 3. Descartes, R. Geometry Selected Works. P. Fermat and Descartes Correspondence, 2nd ed.; Translated from French; URSS: Moscow, Russia, 2010. (In Russian) 4. Galsworthy, J. The Forsyte Saga; Tauchnitz: Leipzig, Germany, 1926; p. 328. 5. Besant, W.H. Conic Sections, Treated Geometrically; George Bell and Sons: London, UK, 1907; pp. 50–86. 6. Levi, H.; Coxeter, H.S.M. Introduction to Geometry. J. Philos. 1963, 60, 19. [CrossRef] 7. Ochkov, V.F.; Fal’koni, A.D. Computer science, algebra, geometry: Four arithmetic curves with pokemon. Inform. V Shkole (Comput. Sci. Sch.) 2016, 9, 57–61. (In Russian) 8. Ochkov, V.F.; Fal’koni, A.D. Seven computing curves or Apollonio bike. Cloud Sci. 2016, 3, 397–418. (In Russian) 9. Ochkov, V.F.; Ochkova, N.A. The project of the monument to three mathematicians or MatMetry. Cloud Sci. 2017, 4, 548–571. (In Russian) 10. Akopyan, A.V.; Zaslavskiy, A.A. Geometric Properties of Curves of the Second Order; MtsNMO Publishing: Moscow, Russia, 2017. (In Russian) 11. Ochkov, V.; Kalova, J.; Nikulchev, E. Optimized Fractal or FMI. Cloud Sci. 2015, 2, 544–561. (In Russian) 12. Ziatdinov, R.; Yoshida, N.; Kim, T. Analytic parametric equations of log-aesthetic curves in terms of incomplete gamma functions. Comput. Aided Geom. Des. 2012, 29, 129–140. [CrossRef] 13. Yoshida, N.; Saito, T. Interactive Aesthetic Curve Segments. Vis. Comput. 2006, 22, 896–905. [CrossRef]

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