Euclidean geometry is "the study of flat space," and named after the mathematician Euclid who lived in 330 B.C. Some major concepts of Euclidean geometry are that "the shortest distance between two points is one unique straight line" and "the sum of the angles in any triangle equals 180 degrees." Euclid also wrote about the parallel postulate, which, if put very simply, states that a single point outside an infinite line can only have one line through it, and those lines will not meet. Non-Euclidean geometry on the other hand, deals with lines that do meet. Two examples of non-Euclidean geometry are Riemannian geometry, which deals with curved surfaces or spheres, and hyperbolic geometry, which studies "saddle-shaped" spaces. In other words, hyperbolic geometry involves spaces that are curved inward like a saddle as opposed to outward, like a sphere.
Projective geometry is a very simple form of geometry that we see in millions of paintings. For example, if you draw two railroad tracks into the distance, the tracks slowly converge at the top of the image. However, the image is not to be taken at face value, but rather what it represents. Although the lines meet in the picture, it intends to portray an image that extends far away. Tom Davis writes at geometer.org that "Today projective geometry is heavily used in a very practical way." For example, he mentions designers use projective geometry to assemble and calculate images on computer screens.
The Encyclopedia Britannica explains that differential geometry studies things like the curvature of certain curves or surfaces. Originally, it drew most of its methods from calculus, which involves adding small parts to determine the whole. Today, however, it relies more heavily on algebra and other related techniques.
Sacred geometry differs from other forms because it has an almost spiritual nature, dating back as far as the ancient Greeks 2500 years ago. It centers around 5 different shapes called the platonic solids. The platonic solids are the tetrahedron, hexahedron, octahedron, dodecahedron, and icosahedron. Scientists saw sacred geometry as pseudoscience until the 1980s when Professor Robert Moon at the University of Chicago revealed that the Periodic Table of the Elements relied on shapes from the platonic solids. Proponents of this field claim that every natural thing, such as DNA, snowflakes, and even the galaxy, follow these geometric patterns.