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Embarking on the journey of **graphing** in **3 dimensions (3D)** is like discovering a new **visual language** that elevates mathematical understanding to a whole new level. This profound tool not only reveals the fascinating relationship among **three variables** but also provides a gateway to explore the depth and complexities of the **physical world** around us.

Whether it’s mapping the **nuances** of a **topographical terrain**, simulating the complex interactions of **variables** in **scientific experiments**, or creating stunning **computer graphics** and animations, **3D graphing** forms the cornerstone of these **endeavors**.

In this article, we’ll demystify the concept of graphing in **3 dimensions**, providing you with **essential insights**, **practical applications**, to understand **3D graphs**.

**Defining Graphing in 3 Dimensions**

**Graphing in three dimensions**, often referred to as **3D graphing**, is a way of representing mathematical functions or sets of data that depend on **three variables**. Instead of plotting points on a two-dimensional plane (like the x and y-axes of a graph), **3D graphing** involves plotting points in **three-dimensional space** along three axes: traditionally labeled as the **x-axis, y-axis, and z-axis**.

In a **3D graph**, each point is determined by three coordinates: (**x**, **y**, **z**), where ‘**x**‘ represents the position along the **x-axis**, ‘**y**‘ represents the position along the **y-axis**, and ‘**z**‘ represents the position along the **z-axis**. These points collectively form a **three-dimensional representation** of the function or data set.

**3D graphing** is commonly used in disciplines such as **physics**, **engineering**, **computer science**, and **economics**, among others, where it allows for a more complete visualization of functions or data that depend on multiple variables.

Below is a generic representation of** 3D** shape.

Figure-1.

**Historical Significance of Graphing in 3 Dimensions**

The history of **graphing** in 3 **dimensions**, or 3D graphing, is intertwined with the history of **geometry**, **algebra**, and **mathematical analysis**.

While the ancient **Greeks** made extensive use of **geometry** in** two dimensions**, the concept of a **third dimension** was not foreign to them. Euclid’s “**Elements**“, dating back to around **300 BCE**, includes definitions and proofs about** three-dimensional figures** such as **cones**, **pyramids**, and **spheres**.

The development of **Cartesian coordinates** by **René Descartes** in the 17th century was a crucial advancement that allowed mathematicians to represent **geometric problems** algebraically and vice versa. Descartes introduced the concept of a **coordinate system**, and while his initial work was in two dimensions, the idea naturally extended to **three dimensions**.

In the **19th century**, significant progress was made in the understanding and visualization of **three-dimensional spaces**. **August Ferdinand Möbius**, a German mathematician and astronomer, made significant contributions in this period, including the discovery of the **Möbius strip**, a two-dimensional surface with only one side when embedded in **three-dimensional space**.

During the same period, mathematicians such as **Carl Friedrich Gauss** and **Bernhard Riemann** developed **differential geometry**, which considers curves and surfaces in three dimensions and beyond. This work laid the groundwork for **Albert Einstein’s general theory of relativity** in the early 20th century.

The **20th century** also saw the development of **computer graphics**, which greatly expanded the possibilities for visualizing functions and data in three dimensions. Today, **3D graphing** is widely used in fields ranging from **mathematics** and **physics** to **computer science**, **engineering**, and **economics**, thanks to software that can easily plot c**omplex surfaces** and data in **three dimensions**.

It should be noted that the history of **3D graphing** is a rich and complex field that touches many areas of mathematics and science, and this summary only touches upon some key developments.

**Properties**

**Graphing** in 3 dimensions (3D graphing) brings several key properties and considerations that distinguish it from graphing in **two dimensions (2D)**. Here are some key properties and aspects to consider:

**Three Axes**

Unlike **2D graphing**, which involves the **x** and **y** axes, **3D graphing** introduces a third axis, typically denoted as **z**. This **third axis** adds a new dimension of depth, allowing you to graph variables that depend on **three inputs** or to represent **three dimensions of data**.

**Coordinate System**

Points in a **3D graph** are identified by **three coordinates** (**x**, **y**, **z**), compared to two in **2D graphing**. These coordinates describe the position of the point in relation to the three axes.

**Orientation and Perspective**

**Orientation** matters a lot in **3D graphing**. Different perspectives can make the same **3D graph** look different, which can sometimes make **3D graphs** more challenging to interpret than **2D graphs**. Modern graphing software often allows users to **rotate and zoom** **3D graphs** to view them from different angles.

**Types of Graphs**

In addition to **3D scatter plots** that represent individual data points in space, **3D graphing** can also involve **surface plots**, which represent a function of two variables, or **contour plots**, which represent three-variable data similar to a **topographical map**.

**Visual Complexity**

**3D graphs** can visually represent more complex relationships than **2D graphs**, including interactions among **three variables** and complex surfaces in **three dimensions**. However, the added complexity also makes **3D graphs** more challenging to create and interpret.

**Data Visualization**

In the field of **data visualization**, **3D graphing** can be used to represent **three-dimensional data**, or **two-dimensional data over time**. However, because **3D graphs** can be harder to interpret, data visualization experts often recommend using **multiple 2D graphs** or other techniques to represent complex data when possible.

**Mathematical Complexity**

The mathematics of **3D graphing** is more complex than that of **2D graphing**, involving **multivariable calculus** and **linear algebra**. These mathematical tools allow for the calculation and representation of **lines, planes, curves, and surfaces** in three dimensions.

Remember that while **3D graphing** can provide **powerful insights and visualizations**, it also comes with challenges in terms of **complexity and interpretation**. Always consider carefully whether **3D graphing** is the best tool for your specific task or whether other representations might be more effective.

## Common 3D Shapes

Three-dimensional (3D) shapes, also known as solids, are figures or spaces that take up three dimensions: length, width, and height. Here are some mathematical examples of 3D shapes, along with their properties:

**Sphere**

A **sphere** is a perfectly symmetrical solid around its center. Each point on the surface of a sphere is an equal distance from its center. A sphere has no **edges** or **vertices**.

**Cube**

A **cube** is a **three-dimensional solid** that has six equal square faces. All the sides and angles are equal. A cube has **12 edges** and **8 vertices**.

**Cylinder**

A **cylinder** has two parallel, congruent bases that are** circular** in shape. The sides of a cylinder are curved, not flat. It has no **vertices**.

**Cone**

A **cone** has a **circular base** and a **vertex**. The sides of a cone are not flat, and they are** curved**.

**Prism**

A** prism** is a **solid** object with two identical ends and all flat faces. The **two ends**, also known as bases, could be in different shapes, including rectangular **(rectangular prism)**, triangular (**triangular prism)**, etc.

**Pyramid**

A **pyramid** is a **3D** shape with a **polygon** as its base and triangular faces that meet at a common **vertex**. The base could be any polygon, such as a square **(square pyramid)** or a triangle **(tetrahedron)**.

**Tetrahedron**

A **tetrahedron** is a pyramid with a** triangular base**, i.e., four equilateral triangles form it. It has **4 faces**, **6 edges**, and **4 vertices**.

**Torus**

A **torus** is shaped like a doughnut. It is a circular ring, where the ring itself also has a circular **cross-section**.

**Dodecahedron**

A **dodecahedron** is a polyhedron with **12 flat faces**. In a regular dodecahedron, these faces are all identical **pentagons**. It has **20 vertices** and **30 edges**.

**Icosahedron**

An** icosahedron** is a polyhedron with **20 faces**. In a regular icosahedron, these faces are all identical **equilateral triangles**. It has **12 vertices** and **30 edges**.

**Applications **

**Graphing in 3 dimensions** (3D graphing) is widely utilized across many fields and disciplines, providing a crucial tool to **visualize** and understand **complex multi-dimensional relationships**. Here are some examples:

**Physics and Engineering**

In **physics**, **3D graphing** is used to represent physical phenomena that depend on **three variables**. For example, electric or gravitational fields in space can be represented as **vector fields** in three dimensions. In **engineering**, it can represent the **stresses** within a structure or the distribution of **temperature** in a system.

**Computer Graphics and Design**

In **computer graphics**, **3D graphing** forms the basis of modeling objects and environments. It helps create detailed models of structures, landscapes, or even entire virtual worlds. In **graphic design**, **3D graphing** is used in the creation of logos, animations, and other graphic elements.

**Geography and Geology**

In **geography** and **geology**, **3D graphing** is used to create **topographical maps and models**, allowing for a detailed representation of the Earth’s surface, including elevations.

**Economics and Finance**

In **economics** and **finance**, **3D graphing** can represent data involving three variables. For example, it can be used to visualize how supply and demand change with price and quantity or to represent a **portfolio’s return, risk**, and **liquidity.**

**Biology and Medicine**

In **biology** and **medicine**, **3D graphing** is used to model and visualize complex structures like proteins or DNA. In medical imaging, technologies like MRI and CT scans use **3D graphing** to create detailed images of the human body.

**Chemistry**

In **chemistry**, **3D graphing** is used to visualize **molecular structures**, which provides insights into chemical properties and reactions. For example, chemists use it to represent electron density clouds around atoms or to show the shapes of molecular orbitals.

**Data Science and Machine Learning**

In **data science**, **3D graphing** can help visualize **multi-dimensional datasets**, aiding in tasks like clustering or outlier detection. In **machine learning**, **3D graphs** can be used to visualize complex decision boundaries or loss landscapes.

**Meteorology**

In **meteorology**, **3D graphing** is used to create **models** and **visualizations** of **weather patterns**, which depend on variables like **temperature**, **pressure**, and **humidity** across three dimensions of space.

Remember that while **3D graphing** is a powerful tool, it’s also important to consider its limitations and challenges. For complex **datasets** or **functions** with more than three variables, other **visualization techniques** might be more appropriate.

**Exercise **

**Example 1**

The function **z = √(x² + y²)**. This represents a cone, extending both upwards and downwards from the origin along the z-axis.

Figuer-2.

**Example 2**

The function **z = sin(x) + cos(y)**. This is a wave-like surface where the height of the waves varies with both x and y.

Figuer-3.

**Example 3**

The function **z = $e^(-x² – y²)$**. This represents a Gaussian or “bell curve” surface, centered at the origin and symmetric in all directions.

Figuer-4.

**Example 4**

The function **z = |x| + |y|**. This forms a pyramid-like shape centered at the origin.

Figuer-5.

*All images were created with GeoGebra.*