Art with Symmetry

Manuel Diaz Regueiro

Art with Symmetry at the Patterns of Beauty Exhibition in Lugo ( June 9-30 , 2026) and in Santiago, in the USC Paraninfo ( November 26 to January 31, 2027)

29 new extremely symmetrical works in 3d

Ambo partial=Ambo faces of a polyhedron.

372paTetrahedronSixCompound1 Ambo partial exact. 6 octahedrons. 3 patterns with 6 reps. NATO

371paTetrahedronSeventyCompound1. Exact partial ambo.70 octahedra. Multiple decagon and enneagon patterns . Turnip basket º

362 pa. TetrahedronFiftyCompound3 Ambo partial exact. 15 octahedra. 15 repetitions of a motif. Medial plane symmetry. cake

57pa Partial Ambo of CubeOctahedronFiveCompound1 Twelve pentagons with valleys and mountains surrounded by 5 twins . But every 3 also form a pattern.

good 16.( Bird). Unknown large bird , or whose name is not known. The vertices are repeated 8 times. The faces 6 times

8 cats with 6 faces?

The vertices are repeated 8 times. The faces 6 times. Cubic symmetry.

354-s60--80cy2. Chameleon 3 pairs of alternating Faces. 6 chameleon vertices. Tetrahedral symmetry.

100-s20-20cy.stl Olympic rings or gentleman with a mustache. 12 patterns of faces of a dodecahedron in rings and 20 Asterix of vertices

sphico3-025. Icosahedral symmetry . Butterflies or Discóbolos in Samothrace.

B398p Three patterns of 1, 3 and 3 repetitions

b344p Unexpected or broken symmetry .6 Squares with swords and 8 trapezoids with daggers . Where is it broken?

b200p. Icosahedral Symmetry . Each vertex and each face with the same pattern. Hidden connections .

b271p. Pentagonal dome . A curious step from symmetry 10 at the base to symmetry 5 at the dome using a pair of patterns that are repeated 5 times on the side.

b21. 2 side patterns that alternate and one above and below . Medial plane symmetry . Four in the face

b261. Incomplete order 6 symmetry . And medial plane. belly

b301. Swords and daggers .6 Squares with 4 swords and 8 trapezoids with 3 daggers .

b82. 2 pattern motifs that are repeated alternately on the side and above and below . And, finally, the most repeated motif, the Trefoil

241. Two patterns that are repeated many times, with 6 and eight sides, but which form a Superpattern 2 by 2 facing each other and in the middle another 2 new ones of 5.

B1 1. Ambiguity Between the sad man and the four leaves. Looking at the work we do not know what the effect or the dominant face is. Well, each one contains the other .

Cuob40cy. Carnivorous plants. Hungry and eager mouths point in all directions in search of food. Symmetry can also indicate danger .

b309 Filigree. There are three dominant patterns , the six- pointed star , the five-pointed star and the decorated triangle or rhombus.

b399. Tetrahedral patterns . The three main motifs appear in space in tetrahedral positions.

332. Lame star . The patterns , the 5-pointed star without one . The square surrounded by curved triangles and the combinations between them.

b323p. Parallels. A central belt of 8 repetitions of the first pattern. 2 belts above and below of the same motif alternating with curved triangles to, at the top, repeat the first pattern again .

159-24m Four- leaf clover . The flowers are missing .

B222. Filigree II. There are three dominant patterns , the six- pointed star , the five-pointed star and the decorated triangle or rhombus.

b410-1. Cubic symmetry. 6 patterns from image 1, 8 from image 2 and with a multiplicity of subpatterns present.

B 415 Taking advantage of a cube. The most important patterns are those of the vertices (8 ), the faces (6) and the edges (12), in addition to other subpatterns present.

B294. Sargadelos. 8 motifs repeating vertically by rotation and a middle plane of symmetry. A 16-pointed star at the zenith and nadir.

In the poster Ambo or (Polyedros Díaz Regueiro ) the Ambo are of the following polyhedra ( some never represented)

The Conway ambo or Kepler rectification is usually exemplified , giving it a simple character, with the cuboctahedron or icosidodecahedron . Here we present 18 examples from the future book 418 ambos of polyhedra, among them many never seen before (Polyedros de Díaz Regueiro ), with properties not known , or rejected even by experts, ( concave -convex ambiguity , for example ) that redefine the general Kepler rectification ).

IcosahedronTenCompound1-CubeFourCompound4-IcosahedronTwoCompound3

RhombicTriacontahedron-GreatTruncatedCuboctahedron

OctahedronThreeCompound3-OctahedronTwoCompound2-OctahedronFiveCompound1

TruncatedIcosahedron-TruncatedPentakisDodecahedron

IcosahedronStellation36-IcositruncatedDodecadodecahedron-PentagrammicAntiprism

TruncatedGreatDodecahedron-RhombicDodecahedronStellation1-EquilateralChamferedIcosahedron

CubeOctahedronFiveCompound1-CubeSixCompound2-IcosahedronTwoCompound1

17 plane symmetry groups

A compassionate mosaic passed to 3d

10 mosaics of Arab tradition and the Alhambra. 10 new ones

A hidden p6 pattern, diamond ring

16 Kolam (l- system )

50 short videos of Menger-Diaz Fractals , polyhedra, both, symmetries and 200 Iranian octagons

Selected works in Bridges for 16 years

All the works of the 16 years in Bridges can be seen with a characteristic and determining profile: 3d symmetry, of equations , geometric , of rotation, of plane symmetry or 3d symmetry. With tools like the l- system or the own laws of 2d or 3d symmetry applied to minimal motifs. And using, exclusively, my own 2d or 3d programs . There are thousands of algorithms, developed over 27 years , so that the symmetries are not only the rotational ones of a glass or a beaker, but polyhedral, objects , motifs or polyhedral patterns that play at making symmetries in 3d, in a way that has never been developed due to its difficulty . Just like the fractal polyhedral patterns , never achieved .

From the 2013 Exhibition and beyond

Making a complete and extensive list of the different posters produced within the Art and Science exhibition, which Igaciencia organized and held at the Lugo Provincial Council from September 19 to November 20 , 2013, is a task to be done calmly over a certain period of time. In this review a month later , December 2013, we want to make a tour , a tasting or a small sample of how much can be said about these Art and Science themes from different points of view: from a Galician ethnomathematical point of view , what Galician constructions can we be proud of and what mathematics they are based on ; of a tool , Geogebra , on which many of the reflections on these constructions that are described are based , of the profiles of many works of art, engineering and architecture that are well -known mathematical functions , such as what could be the symbol of the exhibition, the Eiffel Tower, at the time a symbol of French engineering and today an artistic symbol of Paris, a perfect fusion of art and science in which we can verify with Geogebra that its profile is logarithmic. For all this, detailing that construction with Geogebra is a commitment that needs to be told in these pages .

Art with Symmetry with Geogebra

It is the harmony of the various parts , their symmetry , the happy balance; in a word, it is everything that introduces order, everything that gives unity, which allows us to see clearly and understand at the same time both the whole and the details. Henri Poincaré

We started with a well-known symmetrical figure. We chose a photo of the Taj Mahal.

Which we cut out, with a graphics program, until we get to this other image

What we insert in Geogebra

We press the insert text button in the triangle and move on to the Insert image button.

We position ourselves and click on the point (-4,0) and insert the previous image. We create two points A and B at (4,0) and (4,4) (points on the far right of the image) and draw the line that passes through those two points.

Now we click on this button (Reflect object about the line). Then we point to the image and then to the line AB itself. The result is the demonstration that the Taj Mahal is a perfectly symmetrical figure.

In short, we see in this example how to reconstruct a figure with minimal motifs. For each figure, obtain the minimal motif that through some rotation or symmetry allows us to obtain the complete hexasquel and rosette figure.

Find and draw the minimal motifs of certain logos or artistic figures such as the Mistsubitchi logo, or the design of an Islamic star, analyzing the symmetries or isometries you find and try to decipher a minimal motif that generates the mosaic, a fundamental domain and the base cell.

Rotational symmetry.

 A figure has a center of symmetry when, when performing a rotation or turn (less than a complete turn) around that center, the image produced coincides with the original. This happens with the image of our Celtic symbol triskel. When we perform a 120º rotation of that symbol around its center we verify that the figure coincides with the original and this will happen every time we make a new turn like that. In these cases we say that the figure has rotational symmetry.

A figure has rotational symmetry of order n when that is the number of times it coincides when making a complete turn.

The Triskel has rotational symmetry of order 3. The rose windows of Gothic churches have rotational symmetry of varying order. But let's focus on the Triskel and the Hexaskel.

Our sacred Galician symbols: construction and symmetries.

The Hexasquel. In geogebra, we place the points A(0,0) and B(4,0). We go to regular polygon and draw a regular polygon with 6 sides, starting from points A and B. We draw the segments EB and DA and define the intersection of the 2 segments , G, which will be the center of the hexagon. We define a semicircle passing through G and D. Then we rotate the object around a point according to the angle and indicate the semicircle object and the point G. When it asks us for the angle, we put 60 the first time, and, repeating this process, we put, 120, 180, 240, and 300 and the result will be a hexasquel like those represented in Santa Trega.

The result is a) and without the construction elements it is b)

Buttons we use

We could also construct the six semicircles one by one from the center and the vertices of the hexagon. But in this way , by constructing it with symmetry , we are emphasizing the character of rotational symmetry of order 6 of the figure. If we make the center G visible and draw the circle with center G and passing through one of the points of the hexagon and make the hexagon and all the lines and points formed invisible. We have both the silhouette of a triskelion and that of a hexaskelion .

we mark the points on one of the wings of a triskelion and put the instruction

Polygon[{I,J,L,L,M,N,O,P,Q,R,S,T,U,V,W,Z,A_1,B_1,C_1,D_1,E_1,F_1,G_1,H_1,I_1, J_1,L_1,M_1,N_1,O_1,P_1,Q_1,R_1,S_1,T_1,U_1,V_1,W_1,Z_1,A_2,B_2,C_2,D_2,E_2,F_2,G_2,H_2,I_2, J_2,L_2,M_2,N_2,O_2,P_2,Q_2}]

We change the color and make the points invisible. Then we make G visible, and the center, and rotate the polygon object thus created around that center according to an angle of 120, and 240. The result, an underlined and highlighted triskelion made with rotational symmetry of order 3. We do not show the point G and voilá

We already have our magical and sacred symbol, Galician, in addition to having passed and seen the hexasquel.

The Catenary

Huygens was the first to use the term catenary in a letter to Leibniz in 1690, and David Gregory wrote a treatise on catenary in 1690. If you roll a parabola along a straight line, its focus traces a catenary. Euler also proved in 1744 that the catenary is the curve which, when rotated, gives a minimal surface (the catenoid , the catenary of revolution. The catenoid and the plane are the only surfaces of revolution , which are also minimal surfaces ) .

The catenary also results in a road shape over which a regular polygonal "wheel" can travel smoothly.

The Gateway Arch in St. Louis, Missouri , representing the gateway to the American West, was designed by architect Eero Saarinen as a catenary arch . The Sagrada Familia and other works by Gaudí feature an abundance of catenaries, the curve of a chain held at the ends.

None but the catenary is the figure of a true legitimate arch... And when an arch of any other figure is supported, it is because in its thickness it includes some catenary ( Gregory , 1697) which takes up Hooke 's statement : "the ideal form for an arch is that of an inverted catenary".

The image treated with Geogebra of the Gateway reinforces these ideas of Gregory and Hooke . It should be remembered here how many ancient constructions have only their arches standing, which are millennial and stable because they contain a catenary. It is also the reason that bridges hold up and have initially parabolic shapes. Thus Oldknow showed us a construction of a bridge over the River Cam in England : a parabola of equation y=0.03x ²

And we can show the construction of a bridge in Oporto with Geogebra

The question is , weren't catenaries the stable figures and not parabolas? And the answer may be that a catenary of equation f (x) = a( ℯ ^x + ℯ ^(-x)) / 2 has a first approximation by Taylor's series to the parabola y=ax ² . In fact, even Bernouilli confused the catenary (the curve that a chain makes) with the parabola.

The Eiffel Tower was designed with a logarithmic profile. What didn't you know? y=-3.2 ln(x)-0.7

The springs, since Galileo we know that they draw a parabola in the air. In this case, y=-0.6(x-1.7) ² +3.3. How do we manage to approximate this equation to the image of the photograph? We place the sliders a, b, and k in Geogebra, which we manipulate until we get the formula y=a(xb) ² +k to fit the image of one of the springs. In this manipulation we can observe how the values and even the sign of the three parameters affect the drawn parabola (increasing k raises the parabola, a must be negative to fit a spring, a parabolic fall, b affects moving the parabola to the left or right, ...)

The hyperbola of the cooling towers of the As Pontes thermal power plant 8x ² -2y ² -74x-6y+174=0 It seems incredible that an engineering work has a formula as simple as that and the answer is for two reasons, because it is made with a hyperbolic profile, it is a hyperboloid of revolution, and because in addition the coordinate axes are well chosen so that it does not have xy terms, but they could be chosen better so that they did not have the terms in x and y that it does have. An even simpler equation 8x ² -2y ² =K would remain .

Nautilus . Logarithmic spiral curve. Equation: r=1.24 exp (0.18t)

Tinery in Lugo. Logarithmic spiral curve. Equation: r=4.35 exp (0.17t)

Both in the case of the Nautilus and in this one, the success of the appropriate formula achieved with Geogebra is to mark a point A, which we will be able to move, and define the curve as a Parametric Curve [x(A) + a ℯ ^( bt) cos (t), y(A) + a ℯ ^(bt) sin (t), t, -25.13274, 50]

being a and b sliders that we directly modify ex(A), y(A) coordinates of A, so by moving point A and the sliders a and b we act by visually adjusting the curve to the image that we have or want to approximate.