Question

The equation \(|y / a|^{n}+|x / a|^{n}=1,\) where \(n\) and \(a\) are positive real numbers, defines the family of Lamé curves. Make a complete graph of this function with \(a=1,\) for \(n=\frac{2}{3}, 1,2,3 .\) Describe the progression that you observe as \(n\) increases.

Short answer

Short Answer: As the value of n in Lamé curves increases, the shape of the curve progresses from a rounded diamond shape to a square, followed by a circle, and then a square with increasingly rounded corners.

Step-by-step solution

For \(n=\frac{2}{3}\)

First, we graph the Lamé curve for \(n=\frac{2}{3}\). With \(a=1,\) the equation is: \(|y|^{\frac{2}{3}} + |x|^{\frac{2}{3}} = 1\). To graph this, we will need to consider each of the four possible combinations of signs for \(x\) and \(y\) (since the absolute values are present), and then use a graphing tool/calculator to plot the curve: 1. \(x \geq 0\) and \(y \geq 0\): \(y^{\frac{2}{3}} + x^{\frac{2}{3}} = 1\) 2. \(x < 0\) and \(y \geq 0\): \((-x)^{\frac{2}{3}} + y^{\frac{2}{3}} = 1\) 3. \(x \geq 0\) and \(y < 0\): \(x^{\frac{2}{3}} + (-y)^{\frac{2}{3}} = 1\) 4. \(x < 0\) and \(y < 0\): \((-x)^{\frac{2}{3}} + (-y)^{\frac{2}{3}} = 1\)

For \(n=1\)

Next, we graph the Lamé curve for \(n=1\). With \(a=1,\) the equation is: \(|y|+|x|=1\). We will consider each of the four possible combinations for signs again and plot the curve: 1. \(x \geq 0\) and \(y \geq 0\): \(y + x = 1\) 2. \(x < 0\) and \(y \geq 0\): \((-x) + y = 1\) 3. \(x \geq 0\) and \(y < 0\): \(x + (-y) = 1\) 4. \(x < 0\) and \(y < 0\): \((-x) + (-y) = 1\)

For \(n=2\)

Now, we graph the Lamé curve for \(n=2\). With \(a=1,\) the equation is: \(|y|^2 + |x|^2 = 1\). We will consider each of the four possible combinations for signs again and plot the curve: 1. \(x \geq 0\) and \(y \geq 0\): \(y^2 + x^2 = 1\) 2. \(x < 0\) and \(y \geq 0\): \((-x)^2 + y^2 = 1\) 3. \(x \geq 0\) and \(y < 0\): \(x^2 + (-y)^2 = 1\) 4. \(x < 0\) and \(y < 0\): \((-x)^2 + (-y)^2 = 1\)

For \(n=3\)

Finally, we graph the Lamé curve for \(n=3\). With \(a=1,\) the equation is: \(|y|^3 + |x|^3 = 1\). We will consider each of the four possible combinations for signs again and plot the curve: 1. \(x \geq 0\) and \(y \geq 0\): \(y^3 + x^3 = 1\) 2. \(x < 0\) and \(y \geq 0\): \((-x)^3 + y^3 = 1\) 3. \(x \geq 0\) and \(y < 0\): \(x^3 + (-y)^3 = 1\) 4. \(x < 0\) and \(y < 0\): \((-x)^3 + (-y)^3 = 1\)

Progression Description

Upon analyzing the graphs of the Lamé curves for each of the given values of \(n\), we can make the following observations: 1. For \(n=\frac{2}{3}\), the curve has a rounded diamond shape with concave edges. 2. For \(n=1\), the curve forms a square with vertices at \((\pm 1,0)\) and \((0, \pm 1)\). 3. For \(n=2\), the curve is a circle with a radius of 1. 4. As \(n\) increases further (e.g., \(n=3\)), the curve transforms into a more square-shaped figure with rounded corners. In summary, as \(n\) increases, the Lamé curves progress from a rounded diamond shape to a square, circle, and finally to a square with increasingly rounded corners.

Key concepts

Graphing

Graphing Lamé curves is a fascinating exploration of geometric shapes influenced by changing variables. These curves originate from the equation \(|y/a|^n + |x/a|^n = 1\). When graphing with \(a = 1\), you're observing different scenarios based on the power \(n\).
Each combination of signs for \(x\) and \(y\) requires separate consideration, as the presence of absolute values influences how the functions are plotted.
When plotting these curves using a graphing tool, it's helpful to distinguish between the four quadrants:

• For \(x \geq 0\) and \(y \geq 0\): This quadrant provides the straightforward plot of the equation.
• Negative counterparts are handled similarly: flip the sign of \(x\) or \(y\) as needed.
• The curves manifest symmetrically in all four quadrants.

Graphs of these functions visually demonstrate the beauty and variability of mathematical shapes as they change with different powers.

Equations

The form of the equation \(|y / a|^{n}+|x / a|^{n}=1\) is central to understanding Lamé curves. By setting \(a = 1\), the equation simplifies to \(|y|^n + |x|^n = 1\). The key parameters here are:

• \(n\): Dictates the shape of the curve.
• Absolute values: Ensure calculations get symmetry across both axes.

For example, when \(n = 1\), this translates into the linear equations \(y + x = 1\) and its variants, culminating in a square.
The choice of \(n\) introduces varying degrees of curvature in the Lamé curves. This parameter highlights the transition from diamond-like figures to squares, circles, and rounded squares, illustrating how power functions govern geometric progression.

Progression of Shapes

One intriguing aspect of Lamé curves is their progression as \(n\) varies. As \(n\) increases from values less than 1 to greater numbers, the shape of the curve transforms beautifully:

• For \(n=\frac{2}{3}\): The shape appears as a rounded diamond, with concave edges indicating a porous mixing of x and y influences.
• At \(n=1\): The equation produces a perfect square, with sharp edges.
• Increasing to \(n=2\): The shape becomes a circle, defining perfect equidistance from the center along any axis.
• Finally, as \(n=3\): It reverts to a squarish form, but this time with smoother, more rounded corners.

As \(n\) increases, one can observe how curves initially expand outwards, filling into distinct geometric figures before relaxing into rounded forms. This transformation is a direct representation of mathematical transformations within the framework of Lamé equations.

Power Functions

Power functions, such as those found in Lamé curves, provide powerful insight into how different shapes manifest in coordinate graphs. The equation's exponent, \(n\), plays a pivotal role:

• Each whole number or fraction \(n\) affects the curve's steepness and symmetry.
• Values less than 1 lead to concave forms like the rounded diamond.
• As \(n\) approaches 2, curves become circular, maximizing in symmetry.
• Beyond 2, the form converges into squarish shapes with more subtle edges.

As a fundamental tool, power functions allow mathematicians to predict and model complex variations in geometric shapes, not only offering graphical insight but also a deeper grasp of functional growth and decay. These functions illustrate how increasing complexity in power modifies the resulting shapes beyond simple linear expectations.