Squares In The Real World

[Zulema Estabrooks]

Geometricshapes are everywhere. No matter where you look, almost everything is made up of both two-dimensional (2D) and three-dimensional (3D) geometric shapes. Keep reading for real-life geometric shape examples that make up the world around us.

Two-dimensional shapes are flat figures that have width and height, but no depth. Circles, squares, triangles, and rectangles are all types of 2D geometric shapes. Check out a list of different 2D geometric shapes, along with a description and examples of where you can spot them in everyday life.

Keep in mind that these shapes are all flat figures without depth. That means you can take a picture of these items and you can still determine their shape. The same isn't true for three-dimensional shapes.

Unlike two-dimensional shapes, three-dimensional shapes have width, height and depth. Examples of 3D shapes include pyramids, spheres and cubes. Take a look at these everyday 3D geometric shape examples.

Some of these shapes are interchangeable, of course. For example, a bag might not always be a parallelogram, as there are certainly circular bags and other types possible. This list is also not exhaustive either, as there are many other two-dimensional and three-dimensional geometric shapes.

Spheres are round solid figures. Like circles, they have a radius in the center that is equidistant to every point on the sphere. However, unlike circles, they have volume and depth. Examples of real-life spheres are:

A rectangular prism is a 3D figure where one pair of opposite sides are the same shape, connected by straight, parallel sides. They have four rectangular faces and two square faces. You can find rectangular prisms in these examples:

A three-dimensional figure with one flat side and edges emerging to come together at a point is a pyramid. They can have any shape with three or more sides as their base, including a triangle (triangular pyramid), square (square pyramid) and pentagon (pentagonal pyramid). Examples include:

The world around us is built out of 2D and 3D shapes. Having a basic understanding of geometric shapes and where they can be found in everyday life is a great start to a mathematical education. For more mathematical practice, check out these basic math terms to get you started. You can also take a look at some examples of monomials and polynomials if you're ready to move on to algebraic concepts.

The world is filled with geometric shapes, and geometric forms can be found everywhere. Most things are made up of geometric shapes, whether two-dimensional (2D) or three-dimensional (3D).2 Among the most commonly seen forms around us are squares. You will be surprised by how fascinating and valuable a square can be when you discover more about it. They work well for construction, decoration, and the creation of 3D shapes. The following are a few examples of squares being used in real life:2,3

These popular snacks can be clearly seen to have equal sides on all four, with the opposite sides running parallel to one another. Therefore, these foods can serve as a wonderful illustration of the square objects encountered in daily life.

A square is a perfectly balanced shape with equally long sides. As a result, neither the horizontal nor vertical direction is highlighted. One of the main reasons why photographs and frames are typically square is that it makes the object appear more esthetically pleasing.5

The vast majority of windows installed in homes have a square shape. It is possible for the inner glass of the grilles to be square as well, not just the outer frame. This is because joining two pieces of wood or aluminum with straight cuts is simpler and faster when producing the components in large quantities. Additionally, cutting the glass into square shapes is simpler too.

Squares are easy to make, cut, or build from everyday materials. A square is one of the few regular polygons that can tessellate a surface. This makes a square very practical for construction, decoration, and artistic creation.6

I feel a bit silly because it always bugged me when people asked that in grade school. However, we're both working professionals (I'm a programmer, she's a photographer) and I can't recall ever considering polynomial factoring as a solution to the problem I was solving.

For example, simple trajectory can be modeled with a quadratic function. If you think of time as the input and height as the output, then the positive time for which the polynomial evaluates to zero is precisely the time when the object hits the ground.

For polynomials with integer coefficients the question is roughly the same as "what are the practical applications of algebraic number theory". The usual answers are coding theory and cryptography where factorization (and related operations such as testing whether a polynomial can be factorized) is part of the basic infrastructure from which systems are built or broken. Coding is necessary for digital communication (including telephone, video and satellites) and cryptography has become a basic feature of everyday computer use and commerce.

For polynomials with complex numbers as coefficients the factorization is into linear factors so that factoring is practically the same as numerical root finding (and this is in part true for real numbers as well). Problems in engineering where the location of complex roots of a polynomial determines the behavior of the system are common. For example, stability or instability can be decided by whether all the roots are inside the unit circle, or have positive real part, or other location-based criteria. Oscillations might be periodic if roots are $n$'th roots of $1$ for some $n$, or quasiperiodic behavior if roots are on the unit circle but not all at roots of $1$. A system governed by a partial differential equation would show diffusion (like heat) or wave-like behavior based on the factorization of an associated "differential operator", which is essentially a polynomial.

In general, many phenomena are decomposable into components, pieces or subsystems in a way that (when the systems are modeled mathematically) appears as a multiplicative decomposition of polynomials, with one factor per subsystem.

This is useless. Everything that you study here is completely useless to you later on in life, if you prefer not to study this you can go to a college, or change profession. This university wants you to enrich you with a broader knowledge, either take it or leave it.

Of course, I am lying. Everything that you study can come into use sometimes, often in unexpected places. It is possible that one day number theory will save your life. In the meantime you can just view your studying as a way of learning to do things abstractly.

For example, if I asked you to take out 3 oranges from a pile of 10 oranges. Would this be any different if those were apples? rocks? sheep? bullets? No. It would probably be the same. This level of abstraction is very simple. True.

This problem may seem very different than asking you to buy food for a week with optimal budget (you don't want to spend all your money on groceries, right?), taking into account the weather and how you are likely to spend the following week.

This is a form of abstraction that people are not usually able to do "just like that". Furthermore, even if you do find a general solution, applying it to each problem is again not a trivial matter and is often complicated just as the abstraction part.

Finally, we reach to the point of my babbling above. Mathematics is a wonderful and abstract tool. If you study it, your ability to make the connections between seemingly unrelated problems is likely to get better, your ability to solve the abstract problems is likely to get better, and as a result your ability to solve the problem at hand is likely to get better.

You are a programmer, you need to be able to deal with a lot of problems, they could come in many forms and many ways. You need to be able to see the abstract similarity, and as a good programmer be able to write abstract tools to handle the general problems. Not to rewrite ad-hoc code to solve each problem on its own.

None of the answers so far justify making grade 10 students pointlessly factor polynomials. And for most students, it is indeed a waste of time. Unfortunately, if it were removed from the high school math curriculum, it would be impossible to go on. Now I will tell you why.

Sometimes in life you have to solve a quadratic equation. Not just in school, but in life. It is the basic equation that comes into play when competing factors have to be optimized. You don't always write an equation for these things, but that is what is happening. The classic example is the apple orchard, where you get fewer apples per tree the more you crowd the orchard. The optimum solution is given by solving a quadratic equation.

In real orchards with real apple trees, it is true that the actual equation may not be the simplified quadratic equation of the iconic high school math problem. But the principle of optimization is the same, and it is the quadratic equation which most clearly and in the most simple way illustrates this principle.

Perhaps the most important lesson of high school math is that the physical world can be modelled mathematically, and that mathematical equations have solutions. It is possible to simply write out a formula which solves any quadratic equation but this would be wrong. It obscures the basic idea of what it means to solve an equation mathematically. You cannot begin to explain the general solution of a quadratic equation unless you start with the method of factoring. As pointless as it seems when you are doing it, that is where it leads to and that is why you can't teach math without it.

You need polynomial factoring (or what's the same, root finding) for higher mathematics. For example, when you are looking for the eigenvalues of a matrix, they appear as the roots of a polynomial, the "characteristic equation".

I suspect that none of this will be of any use to someone unless they continue their mathematical education at least to the junior classes like linear algebra (which deals with matrices) and differential equations (where polynomials also appear). And I would also bet that the majority of people who take these classes never end up using them in "real life".

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