Problem 152 - Chemistry

Question

You are asked to help set up a historical display in the park by stacking some cannonballs next to a Revolutionary War cannon. You are told to stack them by starting with a triangle in which each side is composed of four touching cannonballs. You are to continue stacking them until you have a single ball on the top centered over the middle of the triangular base. a. How many cannonballs do you need? b. What type of closest packing is displayed by the cannonballs? c. The four corners of the pyramid of cannonballs form the corners of what type of regular geometric solid?

Step-by-Step Solution

Answer

a. You need 20 cannonballs to form the pyramid. b. The closest packing displayed by the cannonballs is face-centered cubic (FCC) packing. c. The four corners of the pyramid of cannonballs form a regular geometric solid, which is a tetrahedron.

Step 1: a. Calculate the number of cannonballs

To calculate the number of cannonballs, we can count them layer by layer. The base is a triangular shape; as we go up the layers, the number of balls on each side decreases by 1. 1. The base layer has a triangle that has 4 cannonballs on each side, which forms an equilateral triangle. The number of cannonballs in this layer can be represented as the sum of the first four consecutive integers: \(1 + 2 + 3 + 4 = 10\). 2. The second layer also forms a smaller equilateral triangle, with three cannonballs on each side. The number of cannonballs in this layer will be the sum of the first three consecutive integers: \(1 + 2 + 3 = 6\). 3. The third layer will have two cannonballs on each side, forming another equilateral triangle. The number of cannonballs in this layer will be the sum of the first two consecutive integers: \(1 + 2 = 3\). 4. The top layer will have only one cannonball. Now, sum up all the cannonballs in each layer: \(10 + 6 + 3 + 1 = 20\). Therefore, you need 20 cannonballs to form the pyramid.

Step 2: b. Type of closest packing

The arrangement of the cannonballs is in accordance with the closest packing in a regular three-dimensional shape, that is, face-centered cubic (FCC) packing. In FCC, each sphere is in contact with 12 other spheres.

Step 3: c. Type of regular geometric solid

The pyramid's four corners form a regular geometric solid: a tetrahedron. A tetrahedron has four vertices, four equilateral triangular faces, and six edges, each corner touching three spheres in the base and one sphere at the top.

Key Concepts

Geometric Solids

Geometric solids, or three-dimensional shapes, are structures that have length, width, and height. They exist in our world in various forms, like spheres, cubes, and, in our exercise, pyramids. In geometric terms, a pyramid is defined by a polygonal base and triangular faces that converge at a single point called the apex.

In the cannonball stacking problem, the arrangement forms a pyramid. More precisely, it shapes a special kind of geometric solid called a tetrahedron.

• A tetrahedron is a polyhedron with four triangular faces.
• It has four vertices (corners) and six edges.
• All edges and faces of a regular tetrahedron are the same length and shape, making it a regular solid.

Understanding these properties can help visualize how the cannonballs are placed in a three-dimensional space. The structure provides not just a functional stacking form but also a captivating symmetrical shape.

Triangular Numbers

Triangular numbers are a sequence of numbers that can form an equilateral triangle. They tell us how many objects are needed to create a triangle of dots where dots make up the rows of the triangle.

These numbers are crucial in understanding how different layers of cannonballs form. For each layer, the number of cannonballs can be found by calculating the triangular number of the number of sides in the base.

• For the base layer with four sides, the triangular number is calculated as: \(1 + 2 + 3 + 4 = 10\).
• The second layer, with three sides, yields a triangular number: \(1 + 2 + 3 = 6\).
• The third layer with two sides gives: \(1 + 2 = 3\).
• Finally, the top layer is simply 1 cannonball at the apex.

Adding these numbers gives us the total number of cannonballs required, which is 20. This use of triangular numbers provides a clear method to visualize and calculate the stacking.

Close Packing

Close packing is a principle that deals with how spheres can be arranged to take up maximum space with minimal gaps between them. In the context of this problem, it refers to the specific way the cannonballs are stacked to form a stable and dense arrangement.

The cannonballs are organized in a face-centered cubic (FCC) packing.

• In FCC packing, each sphere is surrounded by 12 other spheres, maximizing their contact.
• This method is frequently seen in natural crystals and minimizes space effectively.
• The arrangement results in a uniformly distributed load across the structure, leading to stable stacking.

This packing not only uses space efficiently but also provides stability, which is essential for creating safe and durable arrangements like the pyramid of cannonballs. By understanding FCC packing, we can appreciate both the beauty and functionality of such natural formations.