Friday, August 2, 2019

A Watercolor Retrospective: Part 1.

My first medium of choice was watercolor.  I loved how the different pigments reacted to the paper so I spent a few years experimenting with this, analyzing the pigment qualities and trying to find unique mixtures of the colors. I began to paint “en plein air” during this period, and using a spray bottle I was able to capture the scene in front of me without the paint drying too fast. The moisture of  Woodstock NY didn’t require this but certainly once I reached the dry New Mexico weather, I found the spray bottle useful.

I tried to paint brighter and brighter works and purchased many instruction books on how to obtain deepened color using watercolors. But I began to realize that only with oils could I attain the depth of color that was my goal. Over the years though I still yearned for the feel of watercolor and how as it drops onto the paper unique effects could be obtained.

At first with watercolor I was self taught, buying many books. One of my initial inspirations came from Jeanne Carbonetti’s book (see the photos of references at the end of this long post for other watercolor books that have most  influenced me during thiis watercolor period and beyond):

Some paintings inspired by Carbonette are:

And later I found Georgia O’Keeffe’s watercolors (yes Georgia used watercolors during certain periods).
Georgia O’Keeffe, Evening Star.
I tried copying this or a similar one several times. Unfortunately as a new watercolorist I didn’t realize how important the paper is. Try as hard as I could I seemed not be be able to get the same effects:

In 2002 I visited Santa Fe and spent time at the museums looking at Georgia’s work as well as other New Mexican artists.  One in particular caught my eye: Tom Noble. I love his quirky depictions of New Mexico and his usage of bright colors (I met him in the summer of 2016 at his Taos store and thanked him for his inspiration; he has since died so that I am happy that I met him when I did. Perhaps I will pay homage to him with new paintings of my own now but still capturing his spirit). 
Tom Noble, Guadalupe

For  practice I at this time I copied several of Tom Noble’s  works:

 I more and more wanted to paint full time so I began to consider early retirement from a bigh-stress job in the computer industry managing dozens of computer folk designing, developing, reviewing and testing software for the NYSE traders on the floor.

I found a class at the “Chocolate Factory”  in Red Hook NY in January of 2002 with Betsy Jacaruso, who still teaches - now in Rhinebeck.  Red Hook is just across the river from our vacation home in Woodstock/Red Hook.  So every weekend I painted in Woodstock and during the week I painted in the evenings in Manhattan in the Village.  Here are a few of the paintings from the days with Betsy:

The following was painted “en plein air” at the famous Olana home of Frederick Edwin Church and sold at an auction (Sept. 2003).  I was very excited to see this painting at the top of an article in the Woodstock Times advertising the charity auction:

I still hadn’t retired yet and was trying to paint as much as possible: scenes from the Village as well as from the Woodstock area.

At the first group show of Betsy Jacaruso artists (for me anyway) in Sept. 2013 I sold another painting.  I was so excited! I felt I was on the way to becoming a professional fine artist!! Imagine this painting in the same thin gold frame -with a white mat- as the darker lake scene -which I still have- following this..

Spring Lake, Washington State

During this same period I had to paint my place of work in all its fire and brimstone - and glory:

And finally for this pre-retirement period I painted (several times) the apartment houses and night scene from my 9th street apartment facing North toward the Empire State Building.  I entered it into a juried show at the Woodstock Art Association in the heart of town and it sold!! I wish I had kept this painting as one of my favorites of all times; it reminds me so much of living on 9th Street.  (The recent book about 9th Street Women by Mrs Gabriel which I am reading for the second time - with abstract expressionist artists such as Lee Krasner and Elaine DeKooning- has brought back fond memories of my time in that apartment and my undergraduate and graduate days at NYU at Washington Square, two short blocks away from my last Manhattan apartment.)

This is part I of the watercolor retrospective. Watch this site for the next chapter. Note that in this early part of my career, I was consciously pushing the thought of Mathematics out of my painting experience (or thinking I was).

References: Photos of books I used during my watercolor period, and which I still use as references:

Wednesday, July 31, 2019

Canyon Colores

The canyons of Ghost Ranch and Chama River are an hour’s drive from my Santa Fe home and I try to go there to paint as often as possible. This was the home of Georgia O’Keeffe. Sometimes I try to channel her style but more often I tend to want to use more color and also to use a Palette Knife (or knives).  I like the feel of working with a knife and you can judge for yourself but the results are often better for me.
So, I have done a series of canyon color paintings recently primarily using palette knife.

This is of the first large ones (18x24). I actually took some liberty with a nearby ‘canyon’, Diablo Canyon, just up the road from me. Its rock formation consists of tones of grey rather than the colors of the Ghost Ranch area. But I enhanced the color to reflect the early light of dawn to arrive at this:
Diablo Canyon, Morning Glow, 18x24, oil.
This sold at the Marigold Arts Gallery in Santa Fe.

I then continued with he theme, but this time in my chosen venue of Chama River/Ghost Ranch to paint:
Chama Colores, 12x16, oil.
I reserved this for the annual Las Campanas Charity Auction held at the end of June 2019. I actually painted this as a “concept” painting, having been introduced to the idea by fine artist, Roger Williams.   I chose a few words/terms and constructed a painting base on these: canyon, afternoon light, cliffs, color.

I tried a larger version of this:
Colores, 18x24, oil.

Then this past week, I felt inspired after a painting trip to Abiquiu, and did a small painting with a title Piedra Lumbre I, followed by a larger version, of the cliffs just outside Ghost Ranch:
Piedra Lumbre I, 7x11, oil.

Piedra Lumbre II, 15x24, oil

Piedra Lumbre means Shining Stone.

The Piedra Lumbre paintings have a proportion that is approximately the golden mean, an irrational number, described more fully in other articles on this blog.  I’ve done many paintings, with 10x16 dimension and a fe 15x24, and this 15x24 is 50% larger in each direction, still with a ratio of 10/16 or 5/8. Note that these are two number in the Fibonacci sequence.  1,1,2,3,5,8, .... And as described elsewhere, the radio of two consecutive numbers gets closer and closer to the golden mean as we progress through the sequence.  I think the number is something like 1.610287.... but I will have to look it up.  The ... means continues without end (and cannot be expressed as as a ratio of two whole numbers). Try calculating: 
5/3, 8/5/ 13/8, 21/13,...
A different series, but still a Fibonacci series, might begin with: 1, 3, 4, 7, 11 which actually behaves in a similar fashion.

The golden mean is related tot he golden spiral, a form found in nature everywhere since it in fact represents how growth takes form (think sunflowers, ferns, tree branches, Nautilus Shell). The ratio is thought to be the most pleasing to the eye, perhaps because it is to pervasive and perhaps because it is natural.

But I digress.  I love doing these paintings that I consider somewhat abstract. This allows me to use shapes and colors that appeal to me.

In reality I like the smaller Piedra better and will now analyze why, perhaps touching up the larger version so that it glows in the same way.

By the way, the scene that inspired the Piedra Lumbre pair is this:
Piedra Lumbre Photo

And two more photos:
Piedra Lumbre II palette

Piedra Lumbre II Shadows

The colors mixed on the palette are the ‘grey’ for the shadows (a mix of the Ultramarine Blue and the Vermillion plus white). The remaining colors above the palette I then proceed to mix for variations on the cliffs.  

The full palette: (Cobra unless otherwise stated). Titanium White, Cadmium Yellow Medium, Cadmium Yellow Deep, Vermillion, Madder Lake (alizarin crimson substitute), Ultramarine Blue, Cerulean Blue Hue (really phthalo blue and white), Viridian (W&N Artisas WB Oil), Sap Green, Burnt Sienna and Yellow Ochre.  In front of each pigment I missed a mid-tone of the pigment and white. For these paintings I used very little of the greens.

And finally a photo of the two piedras side by side for size comparison:

Sunday, January 20, 2019

Notes - The Mathematics of Painting - Introduction

What does Mathematics have to do with Art? There's no connection. Right!?? But isn't mathematics all around us in the environment?  I would say that the answer is a resounding "YES!"! Mathematics underlies the patterns found everywhere in nature.

Well I began a new career in art abut 17 years ago, after having worked as a College Professor of Mathematics, a computer programmer and a computer software executive on Wall Street. After 9/11,  I began to question the direction of my life. Yes, I loved working with computers often into the wee hours of the morning, but was there something else?

I thought back at my life and remembered the joy I felt while painting. Perhaps it was time to begin again. I began with "Paint by Numbers" of my childhood but my son persuaded me to try 'real' paints. So I found books on watercolor painting and bought some watercolors and fell in love - and the rest is history.

A funny thing happened in my journey to becoming an artist and landscape oil painter. I noticed how often the 'bibles' of painting included chapters on design and what I perceived as mathematical elements: aerial perspective, 2-dimensional from 3-dimensional, proportions, unequal measures, patterns and motif, self-similarity, informal sub-division, symmetry, dynamic symmetry and the golden mean.  Then terms like fractals, mandelbrot, self-similarity and fibonacci series began to crop up everywhere.

So almost 20 years after beginning my painting journey I realize that it has brought me full-circle back to my roots in mathematics. And my paintings are developing further as I embrace this essential part of my life.  I now actively seek mathematical patterns for my art. After all,  landscape painting is based on what we see around us and mathematics underlies all of nature.


Aerial Perspective:


Golden rectangle:

Informal division:

Dynamic symmetry:

Fractals and chaos:

Notes - Painting on the Left Side of the Brain

Everyone refers to Betty Edwards' book, Drawing on the Right side of the Brain.  And I have to admit that it was an important book for me when I began to draw and paint seriously, in fact, so much so that it contributed to my efforts NOT to use the left side of my brain.

But I was a Mathematician first and throughout my painting career I kept being bombarded with left-side ideas or logic. I couldn't escape it even if I tried. But I kept pursuing my need Not to allow logic to affect my painting.

Finally this year I came to grips with the fact that I am driven by logic; I am an analyst first. And in fact I realized that for me, it's important to accept the logical part in order to grow as a painter.  So I determined that I would not discourage any left-brain ideas popping up while I paint.  And I would marry both sides of my brain perhaps to establish my brand as a painter-mathematician (or mathematician-painter).

We all know that mathematics is at the fundamental core of understanding our universe.  This includes not only theorems and proofs and formulas but the underlying concepts of what makes up everything in nature.  At first glance something like fractals or infinity might be considered ideas that arise out of the right side of the brain. However, ultimately, all the important fundamental ideas of the universe can be boiled down to formulas.  Ah, but you argue, what about the unknowable; how do we explain that.  It turns out that science is getting closer and closer to explaining that which cannot be proven - with theorems and proofs impossible for the layman to follow along with formulas.  It has become the dream of every physicist to discover the underlying essence of the universe with the core theorems describing it.

A century ago a woman mathematicia, Emma Noerther, discovered breakthrough theories that contributed to or in fact drove the development and discovery of major physical concepts.  The book by my friend, Jack Leibowitz,  Hidden Harmony - the Connected Worlds of Physics and Art, describes these so that the layman can appreciate the idea of "Harmony" in our universe and how the understanding of art contributes to this.

But I have deviated from my original thoughts on this matter.  My Mathematical side wants to understand the fundamental theorems of the universe, just as I studied -and taught - the fundamental theorem of calculus.  Let's not forget also that Mathematics lies at the core of Physics.

But back to my original thoughts about painting with the left side of the brain. What does this mean?  To try to clarify this for myself, I am going to list, stream a consciously, any mathematical ideas as I perceive them that I have come across while studying and doing painting - from the simplest concepts to the more complex:

shapes - geometry
composition - balance, unequal measures
informal subdivision
golden rectangle and beauty
golden ratios with the golden mean as the limit
aerial perspective, single point etc.
2- and 3-dimensional shapes in the landscape
light and volume
3-d to 2-d
ruler, verticals,  horizon line,

fractals and Mandelbrot - in the landscape

chaos theory, infinite sereiss
groups and their graphs
infinite combinatorial group theory
symplectic groups (subject of my PhD thesis)

transformations of the plane
tessellations of the plane
4-color mapping theorem
repeated patterns

tools: rulers, calipers, divisions of a rectangle into interesting geometric shapes

And now to another logical area of painting:

color and color theory:

red, blue and yellow and white
CYMB - cyan, yellow, magenta and black
complements:  yellow/purple, red/green, blue/orange

color wheel, harmonic paintings

limited palette, expanded palette

color charts

cooler/warmer; lighter/darker

value-scale (4 values, 10 values)

formulas, integrals, derivatives: fundamental theorem of calculus and how integrals lead to derivatives and vice versa. Meaning of 2nd and 3rd derivatives and how do this relate to painting and art?: Curves, maxima and minima. Mountain shapes, saddles, parabolas, asymptotes.

I have almost forgotten all the mathematics I knew. I hope that the fundamentals are still with me so that if I needed to, I could construct proofs of theorems for example. Or solve differential equations.

So I look at a mountain range and I see a "saddle". And I ask myself, what it its formula (answer later).  I might see a parabola. Well I know this: y = x squared is an example. hyperbola?  circle (y - pi times radius squared, where pi is approximately 3.141257...., not a rational number), hypotenuse of a triangle (x squared plus y squared = hypotenuse squared)
irrational numbers

golden mean ~ 1.612....... not a rational number
fibonacci series (with base of 1: 1, 2, 3, 5, 8, 13, 21, 34)  So a series of rectangles with sides the successive numbers of this series might lead to rectangle that are pleasing to the eye:
3x5 or 6x10;
5x8 or 10x16 or 15x24 or 20x32,;
8x13 or 16x26;
13x21 or 26x42., etcetera.

 So I will use these proportions for my paintings (not counting the frame).

Or if I have a 3 inch wide frame perhaps the panel could be 20x36 for example so that the total framed piece is 26x42.

The ratios of these successive fibonacci numbers get closer and closer to the golden mean:

It's clear from this that we can use any of these ratios to approximate the golden ratio to a degree that we are not able to see any distinction.

So I decide to concentrate initially  on the 5x8 proportion (0.625) , which is the same as 10x16, 15x24 or 20x32.

But I like the idea of using a proportion as close to the golden mean 0.618.... as possible. Well I don't want to work on a panel/canvas as large as 21x34 and I don't want to work in a fractional width or height. And the ratios from 13:21 on all involve fractions or large numbers.

But the 13x21 ratio is very close since 13/21 - 0.61905.  The 8x13 is a little small.  So I start thinking about which ratios are closest to the 0.618 with integral sides that are manageable for me as a painter.

Assume that the largest size canvas I want to use will be something like 36 or 40, which is really not my favorite sizes.   I might try a 21 x 34.
The following seem to be reasonable sizes
580.625 y
8130.615 y
10160.625 y
13210.619 y
21340.618closest to 0.618

So i will order or make panels that are:
fibonacci series: 5x8, 8x13, 13x21, 21x34
(0.625):  10x16, 15x24, 20x32
(0.61538):  16x26

Next: proportions involving radicals; e.g., square root of two, three or five....

Wednesday, January 9, 2019

Karen Halbert: Artist/Mathematician. Defining Moments.

Defining moments:
January 2018: Realized that I needed to return to my second love, Mathematics to explore the connections between it and the passion of the past two decades: art and painting. Adapted the 'brand': Radical Impressionist: A Mathematician Paints. Planning exhibitions.
May 2016. Return to the Rivers, exhibition at the Marigold Arts Gallery.  My son Joshua met a world renowned author and chef, Vikas , at a major political event.  One of his books is titled “Return to the River”. While reading the book into the middle of the night I realized that I have always been drawn to rivers (and oceans).  I was very moved by Khanna’s activities, supporting women and children throughout the world. He has raised himself up from poverty to become beloved by the Dalai Lama, Obama, Hillary Clinton and now my son and by millions watching him on his cooking programs. Khanna also donates dinners to womens’ shelters. Meeting him has opened up a new world for Joshua. I hope to record part of this world in my art. Hence the title of my new exhibit at the Marigold Arts Gallery.
February 2016. While teaching an oil painting class, I found that digging deeper into the techniques of painting and the theory helped me beter undersand what makes a great painting. I have tried to utilize my findings in my new works.
October 2005. Moved to Santa Fe to return to my western roots and to paint the wondrous New Mexico light and landscape.
October 2003. Retired early so that I could dedicate myself to a new career painting full-time. Moved from Manhattan to our summer home in Woodstock, NY to be surrounded by the beauty of the landscape in another artist-rich community.  I was especially inspired by scenes of the Hudson River and the painters of the Hudson River School legacy.
October 2001. Attended an executive program in San Diego, flying there in spite of security concerns. A question asked was: what did I do besides work.  This made me consider my life. It was then that I reminded myself how much art meant ot me when I was younger and how much I ued it for examples in my Mathematics’ classes.  I also realized that an important part of my career involved the presentation of data. I began to teach myself watercolor.
September 11, 2001. At the NYSE I managed the turnover of the software systems to support the opening of the NYSE on Monday, September 17. Then we implemented the system to be highly compliant with Security measures. This was a life-changing period.
1982. Changed from the academic world to the corporate world, transitioning through a start-up company.
1960-64. Chose careers in Mathematics over Art as more practical, though both fields are passions. I took many art courses (and religion and philosophy) as well as the requisite mathematics and physics courses.
1973. Only child, Joshua, born. He attended excellent schools (Fieldston in Riverdale, NY and Carnegie Mellon), studied to be an architect and is an executive for a large Manhattan architectural firm designing and promoting web-based systems; he’s become a computer ‘nerd’ like his mom while his mom became an art ‘nerd’. Josh designed my website,, so that I can update it as needed (often). We are working jointly on a project to upgrade the website to a state-of-the art implementation. I enjoy visiting him and his young family, now in San Diego, frequently, tying the visits into California coast painting trips.
1966. Joined the Peace Corps, leaving a corporate job in computers. Teaching there transformed what I wanted to do. I decided to continue in graduate studies afterwards so that I could teach Mathematics at the undergraduate level.
1955. Moved to Honolulu from Seatlle. I became totally immersed in the Hawaiian culture, while working within an academic challenging environment; the high school I attended was the last ‘english standard’ class in Honolulu. To get into the school required a test. Even today I enjoy returning to one of my second ‘homes’ to paint and absorb the beautiful landscape. My high school friends opened my eyes to a world in which prejudice exists; I vowed to spend my life working to correct this and to trying to find beauty around us and peace.

Early childhood. The excellent junior high school I attended in Seattle opened up the world for me, showing me that I could escape a life of poverty and pursue any dreams I might have. Even today I enjoy returning to my childhood home to visit my close relatives and to view the beautiful scenery and to contemplate painting it (as I did with watercolors ten years ago).

Karen Halbert: Artist/Mathemaician Curriculum Vitae

Curriculum vitae:
2016 – current. Marigold Arts Gallery, Canyon Road, Santa Fe.
2010-2018. Purple Sage Gallery, Oldtown, Albuquerque.
2005-current. Moved to Santa Fe from Woodstock to study with oil painting masters, Roger Williams, Doug Higgins, Bill Gallen and Albert Handell (and others). Volunteer Website Administrator for the Plein Air Painters of New Mexico. Continued to participate in juried exhibitions.  Frequently take painting trips to places such as the Grand Canyon, Hawaii and the Tetons. But I concentrate on painting trips to Ghost Ranch and Abiquiu and to Taos as well as  day trips around the environs of Santa Fe.
2003-2005. Studied with master oil painter, Keith Gunderson. Woodstock, NY.
2003. Retired and Moved full-time to Woodstock, NY.
2002-2005. Studied with watercolorist, Betsy Jacaruso, Woodstock, NY.
December 2001. Began serious training to become an artist.
2001 – 2003. SIAC/NYSE Vice President in charge of Trading Engines, managing over 100 people.
1997-2001. SIAC/NYSE. Managing Director of the Display Book, the premier application used by the Specialists on the floor of the NYSE.
1996-1997. SIAC/NYSE Computer Scientist/Manager of a New Display Book, managing dozens of consultants to update the systems on the NYSE.
1993-1995. SIAC/NYSE. Manager of the Specialists’ Display Book. 
1986-1992. SIAC/NYSE. Computer Programming Consultant.
1983-1986. Philon, Compiler Design Computer Software Startup Company. Programmer and Manager.
1973-1982. College of Mount Saint Vincent, Associate Professor of Mathematics and Computer Science, Riverdale, New York. Taught Beginner classes with references to Artistic terms such as Groups and their Graphs and the Fibonnacci Series. Taught advanced classes in Abstract Algebra with references to Geometric transformations between structures (groups) and to patterns such as tessellations of the plane. Courses taught included Calculus, Complex Analysis and Abstract Algebra as well as introductory courses in Mathematics and Computers to non-math majors.
1969-1973. Graduate Program in Mathematics, Courant Institute of Mathematical Sciences, NYU. PhD in Combinatorial Group Theory. Dissertation. Symplectic Groups (Geometric Transformations of Hyper-space.) Taught in a teachers’ training program and taught undergraduate courses at NYU while a graduate student.
1967-1969. US Peace Corps. Chimbote, Peru. La Escuela Normal Marianista de Chimbote. Taught ’New Math’ to Elementary School teachers in Spanish.
1964-1966. Computer Programmer. Columbia University, Underwater Acoustics Research Laboratory in Dobbs Ferry, NY.
1960-1964. Undergraduate program at NYU majoring in Mathematics. Full NYU scholarship and NY State Regents Scholarship.
1956-1960. High School. Roosevelt High School, Honolulu, Hawaii. National Merit Finalist. Art Award.
1950-1955. Seattle, Washington. Attended Elementary School in Ballard and Catherine Blaine Junior High School in Magnolia.
1942-1950. Vancouver,Washington. Oldest of six children.

Selected Exhibitions: Karen Halbert Artist/Mathematician

Selected Exhibitions – Karen Halbert

November 2018. 10th National National Juried Members Show, Sorrel Sky Gallery. Plein Air Painters of New Mexico, Santa Fe.

June 2018. Santa Fe Plein Aire Fiesta, Juried Paintout and Exhibition. Sorrel Sky Gallery, Plein Air Painters of New Mexico, Santa Fe.

November 2017. 9th National National Juried Members Show, Sorrel Sky Gallery. Plein Air Painters of New Mexico, Santa Fe.

November 2016. 8th National National Juried Members Show, Sorrel Sky Gallery. Plein Air Painters of New Mexico, Santa Fe.

September 2016. Mary Williams Fine Arts Gallery, Juried Exhibition, Plein Air Painters of Colorado, Boulder, Colorado. Award of Merit for "Santa Fe River Turbulence", 10x16, oil.

May 2016. Marigold Arts Gallery, Return to the Rivers, Canyon Road, Santa Fe, NM.

December 2015 – January 2016. Purple Sage Gallery, Solo Show, Hidden Harmony, Oldtown, Albuquerque, NM.

October 2015. Art Collectors’ Gallery, Juried Exhibition, Plein Air Painters of New Mexico (PAPNM) Annual Members Show, Santa Fe, NM

September 2015. Biologique Juried Exhibition, Albuquerque. An exhibition portraying the connections between art and mathematics.

August 2015. Mary Williams Fine Arts Gallery, Juried Exhibition, Plein Air Painters of Colorado, Boulder, Colorado. 

June 2015. Inart Gallery Juried Exhibition, PAPNM Santa Fe Plein Air Festival

May 2014. Featured Artist, Purple Sage Galeria, Trekking through the Land of Enchantment, Old Town, Albuquerque.

June 13 – July 6, 2014. InArt Gallery Juried Exhibition, PAPNM Annual Plein Air Santa Fe (PASF) Paintout, Santa Fe, NM.

May 2-17, 2014. Gary Kim Gallery Juried Exhibition, PAPNM Annual Members Show, Santa Fe, NM.

October 2013 Millicent Rogers Museum Juried Exhibition, PAPNM Annual Members Show, Taos, NM

June 2013. Gary Kim Gallery, Juried Exhibition, PAPNM Annual Plein Air Santa Fe Paintout, Santa Fe.

May 2012 – current. Purple Sage Galeria, Oldtown, Albuquerque, NM

June 2012. Juried Exhibition, PAPNM Annual Paintout, Ruidoso, NM.

November 2011. Placitas Artists Series 4-Person Juried Exhibition, Placitas, NM

October 2011. Millicent Rogers Museum PAPNM Juried Exhibition, Land and Light, Taos, NM

October 2010. Millicent Rogers Museum PAPNM Juried Exhibition, Spirit of Place, Taos, NM

October 2009. Open Space Juried Exhibition, Little Gems, Albuquerque.

August 2009. Wilder Nightingale Gallery Juried Exhibition, PAPNM Annual Members Show, Taos, NM.

June 2004-2006, Betsy Jacaruso Gallery Exhibition: Luminous Visions in Landscape and Still Life, Red Hook, NY

October 2005. Woodstock Art Association Juried Exhibition, The Beat Goes On, Woodstock, NY.

April 2005. Woodstock Art Association Juried Exhibition, Still Life, Woodstock, NY