In the previous post, we have talked about points and their importance in the arts, mathematics, and computer science. We have also mentioned that in Geometry, a point is one of the undefined terms.

Aside from points, there are two more undefined terms in Geometry. They are lines and planes. A line may be drawn through two points, while three points are needed to determine a plane. The representations of these undefined terms are the building blocks of Euclidean geometry.  They can be combined to create shapes, drawings, and sketches such as the painting shown in the first figure. Looking at the painting makes us realize that almost all the things around us are  mostly geometric shapes.

Unitled (1987), Bruce Cohen

In the painting above, we can easily see geometric shapes such as rectangles, triangles, trapezoids (trapezia), and parallelograms. We can also see curves or arcs in vases, flowers, and fruits.  Notice that although the painting seems to be only made by these shapes, the artist has employed a technique to make it look very realistic. For example, the window frames located at the left side of the painting are of the same size, but the artist made the ‘nearer’ frame larger to give a somewhat three dimensional effect.  In doing this, the painter considered the distance of the window frames from the observer.  The farther the frame, the smaller its size on the painting. Observe that this technique is more apparent in the painting by Vincent Van Gogh in the second figure. Continue reading →

Points are building blocks of geometry. If we can remember, a point was one of those “undefined terms” that we have talked about in school mathematics.  Although undefined, our teachers defined it anyway.  We remember them saying that a point is “that which has no length, width, or height.” We also remember that a point is represented with (x,y), where x is the horizontal and y is the vertical; and, that in maps, a point represents a location.

Points are very important especially for painters who use the technique called pointillism. In pointillism, paintings are created meticulously using dots to form figures and sketches.  The painting above titled A Sunday Afternoon on the Island of La Grande Jatte (1884-1886) by Georges Seurat is an example of a painting where pointillism was used. The painting’s dimensions are approximately 2 by 3 meters and it was created painstakingly for two years. Continue reading →

Many people have always viewed mathematics as a difficult subject. Many of them think that mathematics is for the geeks only.  If you are a mathematics teacher, you may have heard your students whispering that getting the root of a cubic function is stupid because it has no use.

Mathematics is a beautiful subject (you will know this as you read more posts in the near future). It has numerous applications in different areas of study including economics, engineering, biology, physics, law, and almost in every pure, applied, and social sciences that we can think of.  In addition, it is also useful in daily life.  We use mathematics to count, to choose the cheapest groceries, to decide when to pay our bills, to analyze whether a loan is reasonable or not, and many others.  Above all, mathematics has taught us to think critically: to recognize and generalize patterns, to make inferences and conjectures, to justify claims, to organize and make sense of data, and to reason logically. Continue reading →