## Lines, Planes, and Perspective

Aside from points, as we have discussed in the previous post, the other two undefined terms in Geometry 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 basic geometric shapes.

In the painting above, we can easily see geometric shapes such as rectangles, triangles, trapezoids , and parallelograms. We can also see curves and arcs in vases, flowers, and fruits.  Notice that although the painting seems to be only made by these shapes, the artist has made 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 so, the painter considered the distance of the window frames from the observer.  The farther the frame, the smaller its size. Observe that this technique is more apparent in the painting by Vincent Van Gogh in the second figure. » Read more

## Points, Pixels, and Pointillism

Points are building blocks of Geometry, it is one of those undefined terms.   If we can recall 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. We are also familiar that points are represented by dots.

If points are important in Geometry, they are also important for painters especially those 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. » Read more

## The Magnificent Magic Cube

If you are fascinated by magic squares, you will be more astonished that magic cubes also exist, one of which is shown in the first figure.

Magic Cubes are cubical arrangements of numbers from $1$ to $n^3$ such that the sums of the rows and the sum of he columns are equal. In the magic cube above, the numbers from $1$ to $3^3=27$ are arranged in a cube and the sum of the numbers on each column and each row is $42$. » Read more

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