Oxido

This is a beautiful, vibrant painting with rich colors. It is abstract, stylized, and simple yet the bright colors and harmonious composition make it seem full of life and energy.   The painting depicts a braid of pure, intense solid colors set against a quiet pastel brown background.   It features a loud but restrained palette and an elegant but sweeping design. 

The composition of the painting guides your eyes in circles, while the multiple gradations of both color and saturation pull your eyes in and out or left and right.   This makes the piece seem to move and shimmer even while it is still on the wall.  

Mathematics

This painting is very geometric.   Each lobe is a segment of a circle, and the canvas is square.   Ignoring color, the painting is screw-symmetric (i.e. rotational but not mirror symmetry).

Color space - You can assign numbers to different colors in a variety of meaningful ways.   One way is to give each color a hue (red-green-blue), saturation (pure tone-gray-white) and brightness (white/pure-black) value.   The colors used in the braid have high brightness and saturation, but explore the whole range of hues.   The background has medium brightness and a lower saturation (it's a mixed tone).

In fact, you can think of each strand as a path and ask, what is a function for the color as I trace along this strand?   This can in turn be used to lift the braid back out of the painting's two dimensions, using hue as a third dimension.   The colors form a repeating pattern that varies as you move outward along a strand, with the outmost color on one strand matching the innermost color on the next lobe.

Knot theory – Knot theory is a deep but accessible region of mathematics that is interesting to laypeople and also the subject of current research.   The shape depicted in this painting appears to be a braid.   We can ask ourselves, which kind of braid is it?   Different people, upon connecting the hidden potions of the path, will argue   that it is

• a single knot with 8 lobes;
• a pair of knots: one with 3 and one with 5 lobes;
• a pair of knots, each with 4 lobes.

Any of these conjectures is justifiable.   In each case you can ask whether the sub knots are simple loops or more complicated knots.   You can also point out that a “color-as-function” idea might let you assign loop connections.

Rapid Reflect

Art

As an artwork, this piece is not very good.   It has an interesting idea and a simple form but the result is clunky and dated.   As you move around the room the pattern shifts and changes, which gives the piece motion.   However its staid composition makes it like a bad movie: it tells a story, but an uninteresting one.  

It uses only black and white colors, and the result is boring.   The rear pattern has five separate regions but I feel they interact jarringly.   A different design choice – to make the regions flow together or to more purposefully contrast – might have respectively given a more pleasing result or a more daring piece.   The lighting is clean, clear and uniform, which is good.   However, the light itself is austere and fluorescent, and has a bluish cast.   There are also construction and installation issues (the casing contrasts badly with the brown walls, and the thick white borders, horizontal slats along the bottom and top of the piece, and the screws in the casing are distracting).

Mathematics

The piece is very geometric, with regions of squares and parallelograms and rectangles.   It has regions of square symmetry.  

If you squint your eyes then highly alternating regions will blur to gray.    This is a form of spatial averaging.

There are two patterns in the piece that interact to give the pattern that we see – the pattern of the slats and the pattern of the background – and each has a characteristic horizontal spacing.   The pattern that we see results from the interaction of these two, and its horizontal spacing is the lowest common multiple of the two original spacings.   (I think more could be done artistically with this idea.)

If you look at the piece edge on, you will see rows of alternating black and white regions, as in the picture below on the left.   You can think of each region as being a binary number, black for 0 and white for 1.   Then the painting represents either a 2-d matrix of binary values, or a sequence of binary numbers, one for each line.   The visual patterns that you see would reflect underlying patterns in the sequence of binary numbers.

One application of binary matrices like these are “cellular automata” (CA).   For example, in the “game of life” CA we impose an initial pattern (usually a 2-d binary matrix) and a rule for finding the next pattern, then iterate that rule.   The patterns evolve in surprisingly interesting ways: you can build patterns that will replicate themselves, move across the field, attack other patterns, fill all space, and more.   An example game-of-life state appears to the right.   CA can also be one-dimensional; the pictures above right show the evolution of a 1-d CA.   The top row shows the initial state, and each row shows   subsequent iterations: time progresses as you read downward in the matrix.

Cellular automata form an interesting area of mathematics that is under current research.   They appear to have deep connections to the understanding of dynamical systems, chaos theory, biological complexity, and genetics.

The construction of this piece requires engineering as well as craftsmanship, and that of course implies mathematics.   Simple arithmetic goes into sizing the pieces and balancing their interactions (the case has to be deep enough to include the lights and wiring, etc).     And of course the advanced materials and the electricity to power the piece are not possible without work done by engineers using calculus and algebra on a daily basis.

Cellular Automota image: S. Wolfram, “Random Sequence Generation by Cellular Automata (1986)”
Game of Life image: Kevin Lindsey, “KevLinDev – ALife” http://www.kevlindev.com/alife/

St Cecilia in Ecstasy

Background

About St. Cecilia (from http://www.ascendedmasters.ac/cecilia.html):

“St Cecilia was a patrician girl of Rome, brought up a Christian, and determined to remain a maiden for the love of God. But her father had other views, and gave her in marriage to a young patrician named Valerian. On the day of the marriage Cecilia sat apart, singing to God in her heart and praying for help in her predicament. When they retired to their room, she said to her husband gently, "I have a secret to tell you. You must know that I have an Angel of God watching over me. If you touch me in the way of marriage he will be angry and you will suffer; but if you respect my maidenhood he will love you as he loves me." "Show me this angel," Valerian replied. "If he be of God, I will refrain as you wish." And Cecilia said, "If you believe in the living and one true God and receive the water of baptism, then you shall see the angel." Valerian agreed and was baptized. Then he returned to Cecilia, and found standing by her side an Angel, who put upon the head of each a chaplet of roses and lilies. ...

Today perhaps Saint Cecilia is most generally known as the patron saint of music and musicians. At her wedding, the acra tell us, while the musicians played, Saint Cecilia sang to the Lord in her heart. In the later middle ages she was represented as actually playing the organ and singing aloud.

Butler, Rev. Alban One Hundred Saints (Bulfinch Press, 1995), page 66

Art

This is a beautiful baroque piece.   The subject is depicted with superbly realistic detail.   The artist subtly abandons realism when it suits his purpose.   He highlights and exaggerates in order to heighten the mood and effect of the painting.   St. Cecilia glows with impossibly glorious light. The important subjects – St. Cecilia, the organ pipes – are drawn with rich, delicate detail and bright colors.   Parts less important to the narrative – the organ body, the background – have markedly less detail, drawing your eye away from them.    

The whole painting effectively and simply serves to draw you in to Cecilia's rapture.   She is bursting with vitality and peace and it is clearly due to the exalted God residing in the heavens of this painting.   St. Cecilia's arms, the focus of her gaze, and the pipes of the organ serve to sweep your point of vision upward through the painting.   Only then do you resolve, in the top left corner, the benevolent angel looking back from the clouds.   It is drawn with moderate detail but similar colors to the heavens.   Like Valerian, it only through St. Cecilia's guidance that the viewer may discover the glory of Heaven. St Cecilia is engaged in playing the organ, an eartly activity, but the whole time she is working through and exalting God.   Her focus is on the spritual, and not the worldly.  

The subject is set against a naturalistic but somewhat irrelevant outdoor background common to most pieces of this time.   The hands are somewhat chubby, especially the fingers on the resting hand.

Mathematics

The composition is structured with triangles that guide your eye upwards.   The proportions of the composition seem to follow the golden ratio.   Her arm and the left of the organ cut off a similar rectangle to the whole piece, and her head and neck again a similar rectangle, leaving St. Cecilia in equal   proportion to the worldly and the heavenly.  

The organ is a rich source of mathematics.   The pipes in the organ contain sound waves   that have exact integer number of peaks and troughs that fit within the tube.   Since there are several ways to fit an integer number of peaks and troughs, there are several different notes coming from the tube. The smallest is the fundamental frequency, giving the pipe its basic tone; the others are the harmonics, coloring the note with overtones.  

The length of the pipe is proportional to the wavelength of the the tone it produces and thus inversely proportional to the freqency of the tone it produces.   The pattern of frequencies in a musical scale is an “add-multiply” pattern: every eight notes represents one octave, doubling the frequency.   Therefore the length of the pipes in the picture above is a graph of a 1/exponential function.   This is why large pipe organs have to be very large: to accommodate the growth rate of the pipes.

Capital & Base of Column

Capital and Base of Column by Hans Sebald Beham of Germany, b. 1500 d. 1550 This work is an engraving (Bartsch 248, Pauli 258) from the Leo Steinberg Collection 2002.788

Art

Though it could serve as an engineering drawing, this is a superbly pleasing artwork as well.   It has balance and simplicity and an elegant classical feel.   The subject is ancient while the style and typography of the engraving are medieval, but the result is timeless. Its subject matter is a Greek/Roman column, an artwork itself with layered details and good proportions.   The engraving has geometric details (cylinders, rings, arches, patterns of chevrons and circles) and natural flourishes (the leaves at the top of the column; the spirals from the crest; the “broken stone” pattern where the column is punctuated).   These contrast harmoniously to make the piece classically pleasing yet organic.

The draftsmanship of this piece is superb.   The work is about 3x1.5” large yet the crosshatching and linework are crisp and precise.   The parchment has aged to a nice sepia which is tastefully highlighted by the matting and simple frame, and contrasts well with the walls.   The piece was framed and matted full-sized to match the surrounding pieces, yet it doesn't get lost – if anything it stands out.

The typography of the caption is ragged but classical and is interesting in itself.

Mathematics

The column consists of a host of interesting geometric objects.   There are spirals, arches (the crown), ellipses, fractals (the broken-stone pattern and the leaves), and patterns composed of circles and lines. The body of the column is not, in fact, a cylinder – the base is wider than the top.   Instead it is probably a section of a very long cone.   The shape of the engraving appears to be a golden ratio; the golder ratio was featured throughout classical architecture and in fact the ornamented section of column just below the crown arches seems to have the golden ratio as well.

The column tapers because the base has to bear more weight than the crown.   The weight increases linearly with height below the crown, while the load-bearing ability will increase as the square of the column area at a given height.   The taper of the column is determined by solving the equations for these constraints simultaneously.   The result happens to be aesthetically pleasing as well: the column appears visually lighter because it tapers away from your eye.

The column displays rotational symmetry about its center axis. There are several periods: the crown repeats on 2 p/4 while the pattern around the base is more like p/36.

The drawing uses perspective techniques to give a realistic rendering; perspective drawing requires extensive geometry.   Notice that the vanishing point lies on the floor: the base is head-on while the crown slants down to the vanishing point.   The circles in the pattern along the base become ellipses when foreshortened.

The crosshatching, when spatially averaged by your eye, appears as light-to-dark shading.