Closeness

In the “real” world you can cover a distance. You can visually tell how close together two points are to each other; even if the points are far away from you, your brain can process the visual information so you can tell the distance between points.

In the truly wonderful and beautiful world of math, these things are not as easily true. Xenos’ Paradox tells us that you can never transverse the distance between points. You can try to get from one point to another point by dividing the distance between the two points in half, and then divide the half closest to the other point in half and so on…to infinity. Essentially, you never reach the other point! There are an infinite number of points between any two points.

On the Cartesian coordinate system, which uses a flat plane, you can calculate the distance between two points and therefore judge the closeness between two points. However, Xeno’s Paradox applies here too.

In mathematical terms, if you try to evaluate the closeness of two points independent of the Cartesian coordinate system which exists only on a flat plane, it is difficult to do. Dividing the distance in half does not work. Imagine two sets of two points out in space where there is not even any type of surface such as a flat plane or the surface of a sphere. How can you tell which points are closest together?

What a great thing to think about!

Greatest Compliment Ever

While I was taking down my three-piece self-portrait, one of the other UVA teachers stopped by to talk to me. She wanted me in her Drawing 2 class in the spring and then to give serious consideration to trying printmaking, her area of specialty. Wow. This is an amazing compliment. I am thrilled! I am giving consideration to perhaps trying for a second BA degree at UVA (or maybe an MA).

Amir Aczel

Ever since reading Descartes' Secret Notebook, I have decided that Amir Aczel is one of my favorite authors. As history is one of my least favorite subjects, this surprises me. Aczel writes his history books as if there is a great mystery being solved by his research. This approach makes his books compelling, even to those who do not like history such as moi! Pick one of his books up and read it today. It will be hours of learning and fun! At the moment I am rapidly reading The Artist and The Mathematician. Awesome book.

Cramer's Rule

I have become fascinated by Cramer's Rule which makes solving systems of linear equations very easy! This has also made me more interested in linear algebra than I have been before. perhaps there are new ways of using matrices out there yet to be discovered. So, now I must decide which math is next.....

Final Display

Gravity, Esh and I all attended the final display for my drawing at UVA, Drawing 1. Gravity was so impressed that he said it did not look like the results of a beginning drawing class, and I agree. The drawings are really good! Here are some pics.

Art 21

The PBS series, Art 21, is great. It focuses on artists of the 21st Century. I have previously made two posts focused on two of the artists, Matthew Richie, and Cai Guo-qiang. The artists talk about their work along with videos of them creating the work and images of various gallery shows/installations of their work. Most of the artists are very articulate and learning what thoughts or feelings are behind their creations. A worthwhile watch.

Yet Another Change

So the final version of spring classes is the Friday drawing class at UVA, a watercolor class from a professional artist, and Calculus A from Stanford U. This should be a nice balance as the two art classes will probably have little out of class work.

Spring Classes

I am going to do 10 credits but chemistry instead of psychology. I am not sure which math yet; either logic or calculus. And, again, the same all day drawing class, but this time Drawing 2.

Spring School

Our community college has increased the number of virtual classes. They now offer psychology classes with labs. I am thinking about taking the second UVA drawing class and a Stanford online math class and a virtual psychology class. It would be a fun mix and 10 credits. I will be busy but I think I will do it.

Alone Time

I love alone time. It is time to revive, a time to grow. Peace, quiet, wonderfulness. Time for thinking. Time for dreaming, relaxing. Time for breathing deeply.

To compensate for my keratoconus, a BIG magnifying glass attached to my light. Absolutely great.

Gnomie preparing to draw at my drawing table!

Cool Site

Following is a cool site about the math behind the show Numb3rs. It has three great short videos generated by Mathematica; especially cool is the Eigenvector Linear Transformation. Enjoy!

http://numb3rs.wolfram.com/507/

Self-Portrait Torso

It was rainy outside so I took photos of the torso portion of my self-portrait on my drawing table. It resulted in glare from the lights so I will retake them when it is sunny.

Rotring vs. Koh-i-noor

About a month ago, I decided to try an additional brand of Rapidograph pen used in technical drawing. In college, I was introduced to Koh-i-noor pens and I used them for years. They produced beautiful lines, but they clog just looking at them. Changing points requires a small wheel which is difficult to even engage onto the pen point and difficult to turn. Along with many other artists, I assumed it was all just part of technical ink drawing.

This all changed when purchased my first .13 mm Rotring pen. Their website states that the pens do not clog. Right. I did not believe it. The claim, however, is true! I have now used their .13 mm for a month and it has not clogged. For any artist who has used Rapidographs, this is nothing short of a miracle! They are also becoming not available in the US! Every artist please stand up and say "We want Rortring!" You will not regret it. No more clogs! Fast point changing! YES!

Lining Up

One of the biggest challenges behind the three piece self-portrait is getting the three pieces to line up and look like they belong. As I have done more and more ink strokes, my stroke style has changed. So now I am having to alter the top of the first piece to match the bottom of the second piece. Interesting challenge.

Wiki Trig

Wikipedia has the most amazing article on trigonometric identities! Wonderful.

Meditation

I know my last post was on what art is, but I must add an addendum. Today, after hours of doing tiny lines, I realized that doing them is meditative. I end up in a sort of trance after awhile, and feel totally brain dead. Then this passes and I feel relaxed and ready to go at it again. Tiny, tiny lines -- thousands of them. Each piece of the portrait is life-size (30" by 22"). In other words, each piece is BIG. Thousands and thousands of little tiny lines.......

What Art Is

I have come across people who think that art is produced through spontaneous creativity. Wrong. Great art is produced through effort -- lots of it. Planning via drawings, preliminary paintings, and much time and effort. Many of the great paintings have taken months, years for the artists to produce. To be an artist requires enormous dedication, physical effort and lots of mental energy.

Bidding

I deepened my understanding of Game Theory tonight during the old Mission Impossible TV show we were watching. Various representatives from various governments were bidding on a nuclear bomb. I realized that someone bidding in order to prevent others from getting the bomb (the MT team) would make sure they had vastly more money than someone else who was bidding in order to process the bomb for their own purposes. The MI team would need to make sure they could out-bid anyone. The other person would have money based on a perceived value worth equation.

What An Angle Is

I like the manner in which an angle is presented in my textbook. The difference between an angle and its measure is clearly delineated. An angle is a formed when one end of a line is rotated around the line' midpoint. The measure is the amount of rotation. Nice.

Clay Witt

From one of UVA's teachers, Clay Witt, this is interesting.

Photoshop CS3

For the first piece of the self-portrait, I used a photograph. It took about 20 hours manipulating the photo (low contrast on the pants and high contrast on background, plus I divided it into nine pieces) to get it the way I wanted it. I used MS Paint and a freebee that come with my computer. I would print one of the nine sections to find out that the registration of objects did not match other pieces that I had already printed. It was a long and frustrating process.

So my husband, bless his wonderful heart, suggested that I download a free thirty day trial of Photoshop. Not doing this earlier in this process was a big, big, BIG mistake. Over a few hours yesterday and today (probably about 8) I have adjusted the contrast on each section of my body but also divided the torso piece and the head piece into the nine pieces. I have printed them all and checked the registration to find that it is perfect with the first printing! Awesome product. Advice to any illustrator or artist who uses photos in their work -- buy Photoshop! It will pay for itself in a very short while.

Exhausting Fridays

My Fridays are exhausting. It usually takes a very short amount of time after I get home to fall asleep in a chair. I get up at 6, begin my drive to C'ville at 7:15, attend class from 9 to 3; and then drive home. I have been getting home around 5.

Self-Portrait Piece 1

This is the first of three pieces of a self-portrait. Hours of work with ink (about 60) plus about 60 hours of planning. Pure joy! The second piece is due at the beginning of November and the third at the beginning of December.

Gary Simmons

I am reading a book by Gary Simmons that is no longer in print but it is really informative. It is called "The Technical Pen". It shows many different types of ink strokes and explains how they are typically used. For my first class project, I now need to decide which stroke s to use and where on the drawing. I love ink. It is amazingly versatile. some sample of Gary Simmons work is below.

Happy

I am glad that I decided to pursue another career even though I am 56. It feels really good to wake up every day and have something driving me to get going. I think it also makes me want to interfere inappropriately in my son's life because I have a fill rich one of my own now. By me pursuing my own goals, we both have great freedom. I am so happy that I have decided to do this.

Homework 1

I really had a lot of fun with homework 1 at my UVA class. We had two weeks to complete the assignment and I sopent around 40 hours. Here is some of the work.

On UVA Video

I (an many others from my class) are pictured in a short video covering the new Studio Art building at UVA, called Ruffin Hall.

http://uvatoday.org/blog/?p=259?19cf86f0

Imaginary i

i is a bautiful number and concept. Today, as I was doing i problems, I realized that all negative numbers could be viewed as "imaginary", or contrived, numbers. Numbers began with the need to count objects. So... how do we count "negative five". We can't; unless we "imagine" five objects that were there, or that will be there, to count. The concept is great.

Perfect Days

It is incredibly hot and humid outside but I am having perfect days. Yesterday, I did math from about 10 AM until 11 PM with very little breaks. It felt wonderful! The weeks before, day after day, I was bent over my drawing table working on a series of ink drawing for my class. I love the combination of the two ways of looking at the world -- one visual and one analytical -- yet the same world with the same patterns. Drawing and math are two different ways to see the world's patterns; and both are equally beautiful! What perfect days I am having right now!

Art 21: Cai Guo-qiang

My husband and I watched an episode of Art 21 that was fascinating. The artist Cai Guo-qiang produces paintings made by exploding gunpowder. He discussed the fact that although he has studied working with gunpowder, he still only so much actual control over what is produced and he loves it. He just lets go and let's the gunpowder produce the art. I love his work. As gunpowder is very ancient to the Chinese culture, I also think it interesting that he has chosen it for his medium.

Matthew Richie

We saw one of the episodes from the Art: 21 PBS series in class today. This one was on an artist named Matthew Richie. It was fascinating. Richie uses mathematical equations and software to generate images that he then turns into massive drawings and sculptures. He uses thoughts about gravity, EM, the weak and strong forces to generate images. Although his approach seemed very different, and a marriage of art and physics seems odd, Leonardo Di Vinci was the same way just tailored to his time period. Following are a couple of articles about Matthew Richie:

http://www.pbs.org/art21/artists/ritchie/index.html

http://www.artonpaper.com/bi/v11n03/column_richie.php

UVA vs. Community College

Today I found out why it is worth it to pay three times the tuition (compared to community college) to attend UVA plus the extra times. The very first class was at least ten times as good. The teacher has been teaching for 20 years and is a full professor there. She has published illustrations and is currently part of a group show in NYC. She is very precise about the materials but very flexible in what we produce as long as it shows effort (and lots of it). Art is a personal statement so her attitude makes sense. I have already spent hours of thinking and working from just a single class! Forget community college and go to a better institution if you can. It will be worth both money and effort; at least it is at UVA which is ranked extremely high.

Intricately Intertwined

In a chapter on symmetry, it was casually mentioned that graphs have not thickness. Of course they don’t but I had never thought about it. In fact, images of parabolas and circles are all wrong because they have a form of line. In reality, they can not have a line but only an infinite number of points that form a curve.

This translated into a thought that stemmed from the book on Cezanne that I recently finished reading. He realized that although people usually begin to draw a ball by first drawing a circle, the actual object has no line at all. So how do I know that the ball exists, and that it is separated from the room around it? It is because of the changes in color, and value. These changes result from the varying surfaces and their reflections of light due to their different curvatures. Everything consists of surfaces, and those surfaces have varying curvatures.

So now we have wrapped back around to mathematics. Different surfaces can be described by a piece of paper and a stick of charcoal. They can also be described by mathematics. Many people think my love of both mathematics and art is an odd combination; but my mind doesn’t. To me they are completely integrated, with their concepts intricately intertwined.

56

I turned 56 about a month ago. Yesterday and today, the middle finger on my left hand has a swollen digit knuckle. It hurts. It is arthritis. I remember the twisted knuckles of my mother and grandmother. Time marches on and we follow our genetic heritage.

New UVA Art Building

Today, I visited the new art building at UVA, Ruffin Hall. It is wonderful. The entire building is one big studio. All rooms have vaulted ceilings, and the entire building has gray concrete floors. Perfect studio! It is near the theater and at the opposite end of the campus from the other classes. I can't wait to begin my new class. It is almost full with 19 students. Only three days from today it will begin.

Exercise

Wow. I need more exercise. I finally got our exercise bicycle in a room with enough space to use the bike. Our bicycle has electronic sensors that read pulse. I was shocked. All I had done was move the bicycle into the room and then sat down to use it and my pulse was 109! Terrible. I used the bike for 10 minutes and the highest my pulse got was 114. The good thing is that I sat for minute after I stopped and my pulse went down to 99 which means that using the bike more often will make a difference -- which is awesome. Exercise bicycle here I come!

Too Much Repetition

My current math class has too many problems that are similar to other problems. For most people, this is so incredibly boring. In fact, I think that is why many people say "I hate math."

So, to make it more enjoyable, I am using software. We have a program that used to be offered by Texas Instruments called Derive. I am also going to begin to use Maple once I upgrade my current software to a Vista compatible version. Learning how to use calculators and various pieces of software is an important part of math that is sometimes over-looked. However, all professionals in physics and math use software as a support to all of their research.

Time Warps

In sci-fi shows, people always remember that time loops are reoccurring and reoccurring. Would this actually happen if time "reset" itself? I think not. We could be looping endlessly, redoing the same moments over and over again and never know.

Assumption of Self-focus

It was the other day that I realized that religions assume that humans have only self-focused motivations. They preach "do this or that for someone else so you can have a good death". They automatically assume that all most humans care about is themselves and their own "position" in their God's eyes when they die or where they will go when they die. I disagree. Many humans do things for others because they are doing something for other people. This is especially true for those individuals who are non-religious. As non-religious individuals, they gain nothing from helping others; no improved afterlife. Could this mean that they are actually more "pure " in their good intentions than people who are devoutly religious? I think so!

For Granted

I have been reading a book (no longer in print so I bought it used), "Techniques of the Impressionists" by Anthea Callen. In the beginning, she points out that paint was not originally available in tubes; it was kept in glass jars with cork or glass stoppers. This somewhat prevented artists from painting outside as paints were not portable.

So Windsor Newton invented paint in tubes. This allowed artists to pack up their oil paints (acrylics had not been invented) and paint out in the open as the Impressionist group often did. http://en.wikipedia.org/wiki/Plein_air I have taken paint in tubes and their portability completely for granted.

Mirrors

Some years ago, I was made aware that some people live surrounded by "mirrors". The mirrors are real to them but others might not be aware of them. Whenever anyone says anything about anything, the mirror people (MP) think that person is talking about them. For MPs, everything around them is about them or for them. That is everything gets reflected onto MPs in MPs minds.

Just the other day I made an additional connection. If MPs try to pickup other people's "signals", chances are they will actually pick up their own not the other person's actual signals. This results in a huge number of self-fulfilling prophecies occurring for them both in their minds and in real life. One end result will be that a MP will think they are extraordinarily good at predicting other people's reactions although this is false as they are actually predicting their own reactions and projecting them on others.

It is important for MPs to become aware of the fact that they have mirrors and that most other people do not have them. If they can realize that they reflect everything back on themselves, they can begin to realize that nay "signals" they pickup from others are more than likely false one, and they should not react to them directly. Instead, it is important for them to confirm or dis-confirm the "signals" they have picked up with the send.

Math Beauty 2

(x^2)' = 2x or 2*x^2-1 or 2x^1

Therefore:

(16)' = (16 * x^0)' = 16 * 0 * x^0-1 = 0

[Yes, x goes to x^-1 but because it is multiplied by 0 it does not matter.]

Therefore:

(2x^2 +2x + 16)' = 4x + 2

Beautiful!

Math Beauty 1

(2^1/2)(2^1/2) = (2)^(1/2+1/2) = 2^1 = 2

Beautiful.

UVA

I am now a part-time student at UVA! The drawing class is from 9 AM to 3 PM on Fridays, but only on Fridays. Should be great to have a different teacher. All of the regular drawing and painting classes at the local community college are taught by the same teacher. He is good but exposure to different ways of doing things is also good. Otherwise it would be like homeschooling where you have exposure to only one teacher class after class after class. Wrong way to learn anything.

Silly Domains

One thing that has always bothered me is the determination of domains in rational expressions before they are simplified. It does not make sense to me and I have not had a logical explanation from anyone yet as to why it is done the way it is done.

So the domain of ((x+6)(x+4))/((x+2)(v+4)) excludes x=-2 and x=-4. But it could just include x=-2 if it were simplified first. Go figure.

Final Class

Yesterday was the final painting class this summer. I will miss it. For the rest of the summer, I will work on our bathroom walls and the front hall, and maybe finish them I also want to finish my other Sargent study.

Next Math Class

Today, I signed up for my net math class with Stanford University Online. Their classes work really well for me. They are introverted and allow for very flexible scheduling. They are also extremely rigorous and I love the work. In my painting class, only four paintings were required and I did six; I just love painting and math and chemistry.

Grass On Beach

Third Stage

I have finished my third stage in my painting of the beach. I have added the grass. Next I have to model the grass so it does not look like flat surfaces.

Disney Woman

Raised in the 50s, Disney animated movies were a big part of my life. The classic Disney woman became a dream of mine; singing in the woods and being surrounded by loving animals.

My dream has become true. This morning, I got out of the shower and two of our cats were sitting there waiting for me. They are so sweet and loving and all they wanted was some petting and affection from me. I have made my childhood goals! How many other people can make that claim?!

Alone Time

A room of your own. Wonderful sentiment by a gifted author (although I hate to admit it but I can't make it through one of Woolf's stream of consciousness books).

I realized one thing I like about math and why I need it in my life. It is introversion in the extreme. Today, I did not go to painting class. Painting is not as introverted as it requires physical activity (my canvases are fairly large) and it is putting self on canvas which is extroverted.

I had to have alone time today. My version of alone time was to sit in the library and watch educational DVDs about painters. It was great. Going to class three times a week, being at home with two people, and periodically being invaded with six teen boys (last night, again) became too much for me; so today, I hibernated. Even the kayak trip around the lake with hubby was introverted because hubby and I went periodically in separate directions and would then meet up for awhile. It was a great day; just what I needed. I feel refreshed and ready to go forward tomorrow.

Tomato Wedges

Lord, I had big chunks of a sweet summer tomato with salt and pepper. Sooooo good. And it has been awhile.

Games People Play

I feel so sorry for my husband. He was raised in a family that knows nothing but how to play games with everyone in their lives. If you have a problem with someone, it is not okay to just say so; instead, you have to do devious things behind their backs. As long as they are nice to your face, they can or say anything behind your back and it is okay to them. So sad. The family is actually built around back-stabbing. Glad that I do not have to have anything to do with any of them.. Very confused bunch. Lying is a way of life.
-------------------------
Actually, I think it all comes from a few of the family members who do not have enough to do with their lives. Please find something to do other than playing games.

Fast Mustard Chicken

You can use any white chicken meat such as breasts or wings. Mix brown mustard with lemon juice and olive oil. Spread half of the mustard mixture in the bottom of a pan then place chicken on top. cover the top of the chicken with the rest of the mustard marinade. refrigerate for at least four hours. Bake chicken at 350 (time will vary with amount of chicken). Serve with wild rice mixture, watercress sprigs, and sliced canned peaches.

12 Hours

So far, I have spent about 12 hours on my beach construction. Today, I finished the layers leading up to the construction of the surfaces (about 6 layers in most of the painting). I still have to put in the grass. Although parts of the painting look disconnected, it is exactly the way I want the surfaces to look. The grass should pull it all together when it is finished. And, I probably have at least 20 more hours on this painting.

Image Construction

The surfaces are almost finished; and when they are I will post a photo.

This afternoon, I realized that I am not painting a painting; but instead, I am constructing an image. An image of dunes, clouds, sky and grass. I can not possibly put into words what it feels like to be doing this 2 dimensional construction. The excitement is wonderful...exhilarating.

GOK Video

Inspiration for anyone. Less than 10 mintes.

http://www.youtube.com/watch?v=BYwKRVJaNEA&eurl=http://video.google.com/videosearch?q=georgia+okeeffe&sitesearch=

Excited

I am really excited over my new painting. I have decided to take pictures of it in stages. Following is a picture of it so far; most of this I did today. I have to still finish the "surfaces" and then paint the geometric grass on top of the surfaces.

UVA For Me

It looks like I will be taking a class at UVA this coming fall. At our local community college, all of the drawing and painting classes during the day are taught by the same person; too much of one perspective even if it is a good one. So I have received permission from the UVA teacher to take a drawing class there. It is a marathon drawing class as it goes from 6 to 3 on Fridays. Should be great!

New Beach Painting Concept

In math, I have always found mapping objects to surfaces fascinating. For instance, on a flat plane, the angles of triangles add to 180 degrees. however, if triangles are mapped to the surface of spheres, the sum of their angles is greater than 360 degrees.

This is the concept I am going to use in my new version of the beach photograph. I am going to create each surface very clearly and well defined and within the confines of distinct geometric objects. then I am going to "map" the grass, also as geometric objects, onto each surface. I am really excited! This approach is a very different one.

True Freedom

I fully realized today that true freedom is a state of mind. I had thought about this before, but today I truly began to understand it. Bounds that we keep on ourselves are in our own minds. Others can influence our lives but we create (or hinder) our own freedom. I am going on with my life independent of my husband and son. Today, I found a new freedom in my painting because I realized that the majority of time I have been doing paintings to please others. Sometimes they please me but not always. I have been allowing others to direct my life far too much. It is my life, not anyone else's. I crave freedom and independence; it is the essence of life itself!

Break Through

During art class, we watched a TV special on Georgia O'Keefe. She was so independent that it was refreshing to watch. At one point, she explained that every summer she left her husband and went to the New Mexico. the interviewer asked if her husband Alfred had any problem with that and she said "He had nothing to say about it. It was something I had to do so I did it." Fabulous! She also said that at some point she realized that she was doing paintings for other people based on others' visions of what she should be doing. She said that the paintings weren't exciting to her so why would they be to anyone else? At was at this point, she decided to start painting what was in her head instead of trying please others.

I realized after that show that I was not enthusiastic over my beach painting at all. I was trying to do a realistic painting of the beach but what I actually saw in my head was the beautiful geometry and texture in the photo. So now I am excited and now I am doing an abstract study based on the beach photo! Bravo.

Offers

For the second time, one of my fellow students asked if I am planning on selling my paintings. Very flattering.

At this point, doing the preliminary sketches is definitely paying off. I did one today of the beach scene with a very turquoise sky and it seems too intense. I think I will use Payne's Gray in the next one, along with Naples Yellow. I'll post a couple when I get them finished.

History Is Laughing

According to the book I am reading on Cezanne, when the Impressionists had their first exhibit, the critics were cruel. One said that the paintings looked like they had been produced by "cross-eyed minds". Another one referred to "Hell and damnation" with reference to the thought that he would look like an Impressionist painting when walking down a Paris street. And one of Monet's water lilies just broke an all time record sale price at $80 million.

New Approach

I mentioned in the last post that I might do preliminary sketches but when I actually began to do it it felt artificial. I need to get used to it. Change.

So I am going to do several sketches on canvas board. Two are going to be two different approaches to the entire painting; one where I work from the furthest away area such as sky to the grasses in the front (I have used this before), and the other will be from overall darkest areas to the middle value areas and then to the lightest (approach my teacher uses). More sketches will involve parts of the painting. I am going to try a different painting technique, too, where I use "sketchy" brush strokes.

I will post photos of my sketches as I go...

Beach Is Next

I have decided to do my next painting from a photo of dunes and beach grasses. I am also going to try a different approach. In the past, I have begun to paint immediately on the canvas. Many professional artists do rough sketches and preliminary small paintings first so I am going to give this a try. In the preliminary paintings, I am going to try different backgrounds

It is so great to constantly learn, change and grow!

When "Private" Isn't Private

When is a private situation not private? When the attacks continue. Stop. I have moved on. You should, too.

Next Painting

My next class assignment was given out today. We are to do an original landscape, self-portrait or painting of a pet. I am going to do a landscape; so I went through old photos tonight. Over the years, I have taken wonderful photos of various things so my selection is terrific. I was going to do a painting from a photo of misty trees and road taken on Skyline Drive but I think I will go with a simpler subject -- either a beach scene or another photo from Skyline Drive showing lots of vegetation. Difficult choices as there are so many great photos in my collection....years of painting ahead, too.

Tonight

After weeks of scorching weather, tonight my son and I went to a shopping center with the top down. The jazz was great and the weather was absolutely yummy. 78 degrees with a light breeze. The clouds were gorgeous. Everyone in the family is in a good mood, too. What a fabulous day this has been.

Study on Sargent's "Poppies"

Flowers on the Deck Painting

I think it is finished. I need to look at it for about a week and then sign it.

My Paintings

I am actually quite pleased with my two current paintings. I am having to adapt my old painting style to one that requires less precise vision but overall, I think they are coming along nicely. I'll post the finished products, hopefully, in a few days!

Cezanne

I have begun to study Cezanne; not just his work but his life. Out of 559 paintings that have been identified as his, only 13 were signed. According to letters he wrote, he felt that many of them needed "further resolution". His work was so individualistic. I found it interesting that he excelled in mathematics and languages but chose to be an artist.

Flowers Painting

I think it will take approximately two weeks to complete my painting of the flowers on the back deck. So far, I am happy with how it is progressing.

In class, we finished the still life and now we are moving one to finding inspiration in other painters' paintings. we are to choose an Impressionist to do a study from. John Singer Sargent did some Impressionist pieces along with his portrait studies so i am choosing one -- "Poppies".

Painting On The Deck

Yesterday, I painted on the back deck. The weather was perfect and I love the flowers back there this year. I can't wait to see what the painting looks like when it is finished.

Painting and Engineering Math

After the long hiatus, doing paintings again feels good. I think I will start going to the studio between classes in addition to during class. I will end up doing extra paintings but that is okay. The teacher is very flexible.

I am also working through an engineering math book , too. Loads of fun and it makes a great review.

Next fall, I think I will take College Chemistry o I can start getting back int0o my biochemical research and the second painting class. It is all so incredibly cheap through the community college and the college itself is only five miles away; not even much gas required.

Angle Formed By Chord and Tangent

Prove that the measure of an angle formed by a chord and a tangent is equal to half the measure of the intercepted arc.

Given circle O, tangent DA where point A is on circle O, and chord BA for circle O. To show that the measure of an angle formed by a chord and a tangent is equal to half the measure of the intercepted arc, I will use proof by cases as the first case has already been proved.

Case 1 is where chord BA is a diameter so point O is on chord BA such that B-O-A. This makes OA and OB radii of circle O so according to Theorem 6.7, tangent DA would be perpendicular to radii OA. This means that the angle measure on both sides is 90 which is indeed, as proved earlier (problem 6.3.2), half of the intercepted arc.

Case 2 is where chord BA is smaller than the diameter. AP1 allows us to construct a line parallel to the tangent DA going through point B, line BC. This creates angle ABC with the intercepted arc, AC. Theorem 6.9 states that the measure of this angle, angle ABC is equal to half of the measure of arc AC.

As lines DA and BC are parallel and secant of chord AB is a transversal, then angles DAB and ABC are alternating interior angles (Definition 4.7) and they are congruent (Theorem 4.27). Therefore, as ABC is half the measure of the intercepted arc, then angle DAB is also half the measure of the intercepted arc; or the measure of an angle formed by a chord and a tangent is equal to half the measure of the intercepted arc.

Proof In Progress

Here is the current proof:

Prove that the measure of an angle formed by a chord and a tangent is equal to half the measure of the intercepted arc.

I will post my proof when I finish it.

Painting Class

I am now in a painting class. So far it is going okay even though my vision has gone to Hell. The next painting is going to be a still life and I am not sure how that will go but I will see. Today was not a good vision day so at one point I looked at my painting and went....oh well. I think for the still life I will have to do some rethinking of my techniques as I will not be able to rely on my vision. As I said earlier, I will find out.

Inscribed Quadrilateral

Prove that if a quadrilateral is inscribed in a circle, then its opposite angles are supplementary.

Assume circle O has an inscribed quadrilateral, ABCD, where four inscribed angles are formed within circle O. In other words, chord AC creates two inscribed angles, ABC and CDB. Each angle equals half the measure of the arc intercepted by the angle (Theorem 6.9). The sum of the measure of the intercepted arcs of the opposite angles is the full circle or 360. That means the angles sum to half of this measure or 180. By Definition 4.14, this means that opposite angles are supplementary.

End

I am coming to the end of my geometry course. I will miss it. I am now on circles, arcs, inscribed angles.

Tangents to a Circle are Congruent

Prove that tangents to a circle from a point are congruent.

Circle O is given. Point P is outside of circle O. In accordance with Definition 6.3, construct segments PA and PB so they form tangents to circle O. Also construct radii AO and BO. Two triangles are now formed – triangle PAB and triangle OAB. As radii are congruent (Theorem 6.1) then AO and BO are congruent. The two radii intersect the two tangent lines perpendicularly (Theorem 6.5). Segment PO then becomes the hypotenuse for triangles PAO and PBO. Segments are also congruent to themselves (AC02) so in both triangles PO is congruent. According to Theorem 4.78, triangles PAO and PBO are congruent as they have congruent hypotenuses and congruent corresponding legs. Theorem 4.20 says that all sides of congruent triangles are congruent; therefore, it can be concluded that PA nd PB are congruent as they are corresponding sides.

Sugar Snap Peas

Tonight was my night to cook (yes, all three of us share making meals, even my 16 year old son). I went through the refrigerator in an attempt to use up remainders before they go bad. The sugar snap pea medley was a success! Snap peas, cherry tomatoes, scallions sauteed in olive oil on high heat. Then I added salt, pepper, capers, and pine nuts. Yum.

Next President

Lord, I hope McCain becomes president. If Obama gets voted in this country might as well place a sign at its door saying "Welcome Terrorist"! And there are already too many of them living here now!

Success

The asparagus was a success!

• Start at least three hours prior to serving so it will be well chilled.
  
• Wash and cut the asparagus into about 1/2 to 3/4 inch pieces.
  
• Boil water and then blanch the aspargus for about 45 seconds to a minute.
  
• Quickly drain asparagus and wash in cold water; place asparagus into a bowl used later for refrigerating.
  
• Add olive oil and lemon juice ; grind pepper on top. Add capers.
  
• Cut beef bratwurst into halves and then into 1/2 inch segments.
  
• Brown bratwurst on high heat in olive oil; then mix with aspargus.
  
• Chill at least three hours and serve.

Risk

I love risk; not reckless risk but well calculated risk. I love going 90 in my car with the top down. I do it on straight stretches when vision is clear and the speed limit is 65 or more. I only do it for short stretches.

I think risk is part of why I keep learning (risk adversity may be the reason why some people stop) and taking classes; forever stretching myself and trying to stretch to my limits. Part of it is risk; risk that I may do well and risk that I might struggle or even fail.

Perhaps that is also why the image of a perfect type of Heaven is revolting to me. There would be no risk and not learning; no growing and changing. I have zero desire to have life too safe. To me, a person might as well be dead as to have a completely safe life where everything is always under control and perfect; even of the perfection is an illusion.

Asparagus

I love asparagus. The other day I decided to try it raw and surprisingly it was yummy. So I am going to try to create an asparagus salad recipe with the asparagus either raw or blanched. I'll keep you posted.

Mountain Climbing

I miss mountain climbing. maybe when Esh is at UVA we can visit him and all go climbing.