#76

Summary

The script details a mathematical exploration of the optimal distribution of toilets in Ocean Park, inspired by the creator's personal experience. Utilizing a Voronoi diagram to analyze the current setup, the creator identifies inefficiencies and attempts to incorporate factors like altitude and walking paths to improve distribution. After recognizing the initial distribution is not optimal, a trial-and-error approach is employed to redistribute toilets, aiming for equal coverage and considering walking time and distance. The conclusion suggests a minimum of 10 toilets for optimal coverage, with a more equal distribution of Voronoi cells, despite some factors like cost-effectiveness and footpath distances being neglected.

Takeaways

• 🎢 The project was inspired by a personal experience at Ocean Park, where the availability of toilets was not as convenient as needed after going through rides.
• 📊 A survey was conducted to understand the average distance people expect a toilet to be from their current location, which was found to be less than a minute's walk or around 80 meters.
• 📐 The mathematical approach involved creating a Voronoi diagram to analyze the distribution of toilets in the park and to determine if it is optimal.
• 📈 The initial Voronoi diagram showed varying areas for each cell, indicating that the distribution of toilets might not be optimal.
• 📉 The script discusses the challenges of incorporating altitude and the winding footpaths into the 2D Voronoi diagram, which would ideally require a 3D model.
• 🚶‍♂️ An attempt was made to account for altitude by generalizing the walking time ratio for uphill and downhill paths, adjusting the midpoints accordingly.
• 🔄 After incorporating altitude, a new Voronoi diagram was created based on walking time, but it still showed unequal areas, suggesting the distribution was not optimal.
• 🔍 The speaker evaluated the distribution by comparing it with the average walking distance people are willing to cover, revealing that some areas had too many toilets close together, while others needed more.
• 🛑 The redistribution process involved creating equations to encircle footpaths and buildings, and partitioning the area into common polygons to calculate the total area and the number of toilets needed.
• 🔢 Through a trial and error method, it was determined that at least 10 toilets were required to fully cover the area without gaps, which was more than the initial 8 toilets.
• 📉 Despite the successful redistribution, the script acknowledges that there were neglected factors such as cost-effectiveness and the actual footpath distances that could affect the accuracy of the Voronoi diagram.

Q & A

What was the inspiration behind the math project on the distribution of toilets in Ocean Park?

The inspiration came from a personal experience during a math trip to Ocean Park, where the presenter found the toilets not readily available despite signs indicating their presence.

What was the average distance people surveyed thought a toilet should be from any point in the park?

The average answer from the survey was less than a minute's walk or around 80 meters from anywhere in the park.

What mathematical tool was used to analyze the distribution of toilets in Ocean Park?

The Voronoi diagram was used to analyze and partition the space based on proximity to the set points, which in this case were the toilets.

How does a Voronoi diagram represent the distribution of toilets?

A Voronoi diagram partitions space into cells, each containing all points closer to one toilet than to any other, indicating equal access distance to that toilet from any point within the cell.

What was the initial conclusion about the distribution of toilets based on the Voronoi diagram?

The initial conclusion was that the distribution was not optimal, as the areas of the Voronoi cells varied significantly, indicating unequal access distances to the toilets.

What factors did the presenter attempt to incorporate into the model to improve its accuracy?

The presenter attempted to incorporate factors such as altitude changes, winding footpaths, and the general walking speed relative to slope steepness.

Why was the 3D model not used to account for altitude changes?

The 3D model was not used because the graphing calculator used, Desmos, is a 2D tool and does not support a Z variable for altitude. Additionally, obtaining the altitudinal changes for the entire park was not feasible.

How did the presenter generalize the effect of altitude on walking time?

The presenter assumed a ratio of 1:1.1 for walking uphill versus downhill, meaning it takes more time to walk uphill than downhill for the same distance.

What method did the presenter use to redistribute the toilets to achieve a more optimal distribution?

The presenter used a trial and error method, creating equations to encircle footpaths and buildings, partitioning the area into polygons, and calculating the number of circles needed to cover the area without overlapping.

What was the minimum number of toilets required to cover the entire area according to the presenter's redistribution trials?

A minimum of 10 toilets was required to fully cover the entire heptagonal area without any gaps.

What factors were neglected in the redistribution process that the presenter acknowledged?

The presenter acknowledged that factors such as the cost-effectiveness of having more toilets and the exact distances of footpaths to the toilets were not accounted for, which could affect the accuracy of the redistributed model.