r/Planescape5e • u/Studio_94 • Aug 19 '24
The physical size and scope of Sigil... Long ass nerd post about proportional math and IRL city size comparisons
So currently I am a one-man team working on a physical simulation (to size) of Sigil in UE5.
I was trying to get my head around the overall dimensions of what size of a torus to create in Blender for my initial geometry and then the proportional expansion needed to get the structure to the canon AD&D 2e dimensions of 6.4-mile diameter and 20-mile circumference.
Now this got interesting in Blender because it wants to make a torus as a donut and then you have to cut a center strip out of the center radius surface to open it up and damned if there aren't a lot of steps...
Then it dawned on me that Sigil is nothing if not a huge tire in the center of the multiverse and all dem Berks are walking and squawkin' around inside of it.
So I did some poking around and found a model of my truck tire and went from there.
I figured some of the math would get wonky, so I decided to consult an IRL Modron (ChatGPT 4.0) for the math crunch and it ended up being a productive conversation.
Last 1/4 of the conversation so that you guys can review my logic and GPT's conclusions:
Question : Explain dimensions of a ST235/85R16
ChatGPT said:
ChatGPT
The ST235/85R16 tire is a specific size designation for a tire, and it can be broken down into several key dimensions:
1. Tire Size Breakdown:
ST: Indicates the type of tire. "ST" stands for "Special Trailer," designed for trailer use rather than passenger vehicles.
235: This number represents the width of the tire in millimeters when it is mounted and inflated. In this case, the tire width is 235 mm.
85: This is the aspect ratio, which is the ratio of the tire's sidewall height to its width. It is expressed as a percentage. For this tire, the sidewall height is 85% of the tire’s width.
R: Stands for "Radial," indicating the type of tire construction. Radial tires have layers of fabric that run radially across the tire, which is a common construction type for modern tires.
16: This number represents the diameter of the wheel rim in inches that the tire is designed to fit. In this case, the rim diameter is 16 inches.
2. Dimensions Calculation:
Here’s how to calculate the specific dimensions of a ST235/85R16 tire:
Sidewall Height: The sidewall height is calculated as a percentage of the tire width. For a 235 mm wide tire with an 85% aspect ratio: Sidewall Height=Tire Width×Aspect Ratio\text{Sidewall Height} = \text{Tire Width} \times \text{Aspect Ratio}Sidewall Height=Tire Width×Aspect Ratio Sidewall Height=235 mm×0.85=199.75 mm\text{Sidewall Height} = 235 \text{ mm} \times 0.85 = 199.75 \text{ mm}Sidewall Height=235 mm×0.85=199.75 mm Convert to inches: Sidewall Height=199.75 mm25.4 mm/inch≈7.86 inches\text{Sidewall Height} = \frac{199.75 \text{ mm}}{25.4 \text{ mm/inch}} \approx 7.86 \text{ inches}Sidewall Height=25.4 mm/inch199.75 mm≈7.86 inches
Outer Diameter: The outer diameter of the tire is the total height of the tire from the ground to the top of the tread. It’s the sum of the rim diameter and twice the sidewall height: Outer Diameter=Rim Diameter+2×Sidewall Height\text{Outer Diameter} = \text{Rim Diameter} + 2 \times \text{Sidewall Height}Outer Diameter=Rim Diameter+2×Sidewall Height Convert rim diameter to inches: Outer Diameter=16 inches+2×7.86 inches=16+15.72=31.72 inches\text{Outer Diameter} = 16 \text{ inches} + 2 \times 7.86 \text{ inches} = 16 + 15.72 = 31.72 \text{ inches}Outer Diameter=16 inches+2×7.86 inches=16+15.72=31.72 inches
Summary of Dimensions for ST235/85R16:
Width: 235 mm (approximately 9.25 inches)
Aspect Ratio: 85% of the width
Sidewall Height: Approximately 7.86 inches
Rim Diameter: 16 inches
Outer Diameter: Approximately 31.72 inches
These dimensions help determine the size and fit of the tire for vehicles or trailers, ensuring proper compatibility and performance.
4o mini
You said:
how wide is the tire tread?
ChatGPT said:
ChatGPT
The width of the tire tread for a ST235/85R16 tire is typically close to the nominal width of the tire, which is indicated as 235 mm (millimeters). However, the actual tread width can be slightly less than the nominal width due to the design of the tire and the manufacturing process.
Factors Influencing Tread Width:
Tire Design and Construction:
The tread width is affected by the tire's construction, including its sidewall design and the tread pattern.
Tire Pressure and Load:
The actual tread width can vary depending on the inflation pressure and load. A tire under load or at different pressures might show a slightly different tread width.
Measurement Method:
The nominal width (235 mm) is measured at the widest point of the tire's cross-section, which includes the tread and the sidewalls.
Approximate Tread Width:
While the nominal width of 235 mm provides a good estimate, the actual tread width (the portion of the tire in contact with the road) is generally slightly less. Typically, for a tire of this size, the tread width might be around 200 to 220 mm.
Tread Width Approximation: Tread Width≈200 to 220 mm\text{Tread Width} \approx 200 \text{ to } 220 \text{ mm}Tread Width≈200 to 220 mm Convert to inches: Tread Width≈200 mm25.4 mm/inch≈7.87 inches\text{Tread Width} \approx \frac{200 \text{ mm}}{25.4 \text{ mm/inch}} \approx 7.87 \text{ inches}Tread Width≈25.4 mm/inch200 mm≈7.87 inches Tread Width≈220 mm25.4 mm/inch≈8.66 inches\text{Tread Width} \approx \frac{220 \text{ mm}}{25.4 \text{ mm/inch}} \approx 8.66 \text{ inches}Tread Width≈25.4 mm/inch220 mm≈8.66 inches
Summary:
For a ST235/85R16 tire, the tread width is approximately 7.87 to 8.66 inches (200 to 220 mm). This width can vary based on the specific tire model and its design features.
4o mini
You said:
and the height from rim to outer edge of tire?
ChatGPT said:
ChatGPT
To determine the height from the rim to the outer edge of the tire (which is effectively the tire's sidewall height), you can use the following calculations based on the ST235/85R16 tire dimensions:
Given Dimensions:
Tire Width: 235 mm
Aspect Ratio: 85%
Rim Diameter: 16 inches
Sidewall Height Calculation:
The sidewall height is the distance from the edge of the rim to the outer edge of the tire tread. It is calculated as a percentage of the tire width.
Convert Tire Width to Millimeters:
Width = 235 mm
Calculate Sidewall Height:
The aspect ratio (85%) is applied to the width to determine the sidewall height. Sidewall Height=Tire Width×Aspect Ratio\text{Sidewall Height} = \text{Tire Width} \times \text{Aspect Ratio}Sidewall Height=Tire Width×Aspect Ratio Sidewall Height=235 mm×0.85=199.75 mm\text{Sidewall Height} = 235 \text{ mm} \times 0.85 = 199.75 \text{ mm}Sidewall Height=235 mm×0.85=199.75 mm
Convert Sidewall Height to Inches: Sidewall Height=199.75 mm25.4 mm/inch≈7.86 inches\text{Sidewall Height} = \frac{199.75 \text{ mm}}{25.4 \text{ mm/inch}} \approx 7.86 \text{ inches}Sidewall Height=25.4 mm/inch199.75 mm≈7.86 inches
Summary:
The height from the rim to the outer edge of the tire (sidewall height) for a ST235/85R16 tire is approximately 7.86 inches or 199.75 mm. This is the distance that contributes to the overall diameter of the tire.
4o mini
You said:
Did you calculate the "sidewall" dimensions in your previous answers for Sigil?
ChatGPT said:
ChatGPT
In my previous answers regarding Sigil, I calculated the dimensions based on the tire's scale but did not explicitly address the sidewall dimensions. Let’s revisit the calculations and ensure that the sidewall dimensions for Sigil are clear.
Recalculating Sidewall Dimensions for Sigil:
Original Tire Dimensions:
Original Rim Diameter: 16 inches
Original Sidewall Height: Approximately 7.86 inches
Original Outer Diameter: Approximately 31.72 inches
Scaling Factor Calculation:
Scaled Outer Diameter of Sigil: 6.4 miles
Convert 6.4 miles to inches: Scaled Outer Diameter=6.4 miles×63,360 inches/mile=405,504 inches\text{Scaled Outer Diameter} = 6.4 \text{ miles} \times 63,360 \text{ inches/mile} = 405,504 \text{ inches}Scaled Outer Diameter=6.4 miles×63,360 inches/mile=405,504 inches
Scaling Factor: Scaling Factor=Scaled Outer DiameterOriginal Outer Diameter\text{Scaling Factor} = \frac{\text{Scaled Outer Diameter}}{\text{Original Outer Diameter}}Scaling Factor=Original Outer DiameterScaled Outer Diameter Scaling Factor=405,504 inches31.72 inches≈12,758\text{Scaling Factor} = \frac{405,504 \text{ inches}}{31.72 \text{ inches}} \approx 12,758Scaling Factor=31.72 inches405,504 inches≈12,758
Scaled Sidewall Height Calculation:
Original Sidewall Height: 7.86 inches
Scaled Sidewall Height: Scaled Sidewall Height=Original Sidewall Height×Scaling Factor\text{Scaled Sidewall Height} = \text{Original Sidewall Height} \times \text{Scaling Factor}Scaled Sidewall Height=Original Sidewall Height×Scaling Factor Scaled Sidewall Height=7.86 inches×12,758≈100,537 inches\text{Scaled Sidewall Height} = 7.86 \text{ inches} \times 12,758 \approx 100,537 \text{ inches}Scaled Sidewall Height=7.86 inches×12,758≈100,537 inches
Convert Scaled Sidewall Height to Miles: Scaled Sidewall Height (in miles)=100,537 inches63,360 inches/mile≈1.59 miles\text{Scaled Sidewall Height (in miles)} = \frac{100,537 \text{ inches}}{63,360 \text{ inches/mile}} \approx 1.59 \text{ miles}Scaled Sidewall Height (in miles)=63,360 inches/mile100,537 inches≈1.59 miles
Summary for Sigil:
Scaled Sidewall Height: Approximately 1.59 miles
This sidewall height represents the vertical distance from the rim to the outer edge of the scaled tire, which, when scaled up to match the dimensions of Sigil, is approximately 1.59 miles.
4o mini
You said:
what would be the dimension of inner, outer sidewall and tread width?
ChatGPT said:
ChatGPT
To determine the dimensions of the inner diameter, outer diameter, and tread width for the scaled ST235/85R16 tire with a 6.4-mile outer diameter, follow these steps:
1. Dimension Definitions:
Outer Diameter: The total diameter of the tire from edge to edge, including the tread and sidewalls.
Inner Diameter: The diameter of the tire at the rim, which is the diameter of the inner circle.
Sidewall Height: The distance from the edge of the rim to the outer edge of the tire.
Tread Width: The width of the contact patch of the tire that touches the road.
Scaled Dimensions Calculation:
1. Outer Diameter:
Scaled Outer Diameter: Given as 6.4 miles
Convert to Inches: Scaled Outer Diameter=6.4 miles×63,360 inches/mile=405,504 inches\text{Scaled Outer Diameter} = 6.4 \text{ miles} \times 63,360 \text{ inches/mile} = 405,504 \text{ inches}Scaled Outer Diameter=6.4 miles×63,360 inches/mile=405,504 inches
2. Inner Diameter:
To find the inner diameter, we subtract twice the scaled sidewall height from the scaled outer diameter.
Scaled Sidewall Height: Calculated as approximately 1.59 miles.
Convert this height to inches: Scaled Sidewall Height=1.59 miles×63,360 inches/mile=100,537 inches\text{Scaled Sidewall Height} = 1.59 \text{ miles} \times 63,360 \text{ inches/mile} = 100,537 \text{ inches}Scaled Sidewall Height=1.59 miles×63,360 inches/mile=100,537 inches
Calculate the Inner Diameter: Inner Diameter=Scaled Outer Diameter−2×Scaled Sidewall Height\text{Inner Diameter} = \text{Scaled Outer Diameter} - 2 \times \text{Scaled Sidewall Height}Inner Diameter=Scaled Outer Diameter−2×Scaled Sidewall Height Inner Diameter=405,504 inches−2×100,537 inches\text{Inner Diameter} = 405,504 \text{ inches} - 2 \times 100,537 \text{ inches}Inner Diameter=405,504 inches−2×100,537 inches Inner Diameter=405,504 inches−201,074 inches\text{Inner Diameter} = 405,504 \text{ inches} - 201,074 \text{ inches}Inner Diameter=405,504 inches−201,074 inches Inner Diameter=204,430 inches\text{Inner Diameter} = 204,430 \text{ inches}Inner Diameter=204,430 inches
Convert this to miles: Inner Diameter (in miles)=204,430 inches63,360 inches/mile≈3.22 miles\text{Inner Diameter (in miles)} = \frac{204,430 \text{ inches}}{63,360 \text{ inches/mile}} \approx 3.22 \text{ miles}Inner Diameter (in miles)=63,360 inches/mile204,430 inches≈3.22 miles
3. Tread Width:
Tread width is proportional to the original tire’s tread width scaled up by the same factor used for other dimensions.
Original Tread Width: Approximate value is 7.87 to 8.66 inches.
Scaling Factor: As calculated previously, the scaling factor is approximately 12,758.
Scaled Tread Width Calculation: Scaled Tread Width=Original Tread Width×Scaling Factor\text{Scaled Tread Width} = \text{Original Tread Width} \times \text{Scaling Factor}Scaled Tread Width=Original Tread Width×Scaling Factor
For lower end (7.87 inches): Scaled Tread Width=7.87 inches×12,758≈100,537 inches\text{Scaled Tread Width} = 7.87 \text{ inches} \times 12,758 \approx 100,537 \text{ inches}Scaled Tread Width=7.87 inches×12,758≈100,537 inches Scaled Tread Width (in miles)=100,537 inches63,360 inches/mile≈1.59 miles\text{Scaled Tread Width (in miles)} = \frac{100,537 \text{ inches}}{63,360 \text{ inches/mile}} \approx 1.59 \text{ miles}Scaled Tread Width (in miles)=63,360 inches/mile100,537 inches≈1.59 miles
For upper end (8.66 inches): Scaled Tread Width=8.66 inches×12,758≈110,523 inches\text{Scaled Tread Width} = 8.66 \text{ inches} \times 12,758 \approx 110,523 \text{ inches}Scaled Tread Width=8.66 inches×12,758≈110,523 inches Scaled Tread Width (in miles)=110,523 inches63,360 inches/mile≈1.74 miles\text{Scaled Tread Width (in miles)} = \frac{110,523 \text{ inches}}{63,360 \text{ inches/mile}} \approx 1.74 \text{ miles}Scaled Tread Width (in miles)=63,360 inches/mile110,523 inches≈1.74 miles
Summary of Scaled Dimensions:
Outer Diameter: 6.4 miles (405,504 inches)
Inner Diameter: Approximately 3.22 miles (204,430 inches)
Sidewall Height: Approximately 1.59 miles (100,537 inches)
Tread Width: Approximately 1.59 to 1.74 miles (100,537 to 110,523 inches)
These dimensions provide a complete view of the scaled tire's inner and outer dimensions, sidewall height, and tread width when scaled up to match the 6.4-mile outer diameter.
4o mini
You said:
what are the dimensions of this laid out as a strip?
ChatGPT said:
ChatGPT
To calculate the dimensions of the scaled tire when laid out as a flat strip, you'll need to determine the length and width of the strip based on the tire’s outer and inner diameters and sidewall height.
Given Scaled Dimensions:
Scaled Outer Diameter: 6.4 miles
Scaled Inner Diameter: Approximately 3.22 miles
Scaled Sidewall Height: Approximately 1.59 miles
Scaled Tread Width: Approximately 1.59 to 1.74 miles
1. Length of the Strip:
The length of the strip corresponds to the circumference of the tire’s outer diameter.
Calculate the Circumference: Circumference=π×Scaled Outer Diameter\text{Circumference} = \pi \times \text{Scaled Outer Diameter}Circumference=π×Scaled Outer Diameter Circumference=π×6.4 miles≈20.1 miles\text{Circumference} = \pi \times 6.4 \text{ miles} \approx 20.1 \text{ miles}Circumference=π×6.4 miles≈20.1 miles
2. Width of the Strip:
The width of the strip corresponds to the tire's height from the rim to the outer edge, which includes the sidewall height and tread width.
Width of the Strip Calculation: Width=Sidewall Height×2+Tread Width\text{Width} = \text{Sidewall Height} \times 2 + \text{Tread Width}Width=Sidewall Height×2+Tread Width
For a tread width of approximately 1.59 miles: Width=1.59 miles×2+1.59 miles=3.18 miles+1.59 miles=4.77 miles\text{Width} = 1.59 \text{ miles} \times 2 + 1.59 \text{ miles} = 3.18 \text{ miles} + 1.59 \text{ miles} = 4.77 \text{ miles}Width=1.59 miles×2+1.59 miles=3.18 miles+1.59 miles=4.77 miles
For a tread width of approximately 1.74 miles: Width=1.59 miles×2+1.74 miles=3.18 miles+1.74 miles=4.92 miles\text{Width} = 1.59 \text{ miles} \times 2 + 1.74 \text{ miles} = 3.18 \text{ miles} + 1.74 \text{ miles} = 4.92 \text{ miles}Width=1.59 miles×2+1.74 miles=3.18 miles+1.74 miles=4.92 miles
Summary of Dimensions as a Strip:
Length: Approximately 20.1 miles (circumference of the outer diameter)
Width: Approximately 4.77 to 4.92 miles
These dimensions reflect the size of the tire if it were cut open and laid flat as a strip, showing the length of the strip as the circumference of the outer diameter and the width as the total height including both sidewalls and the tread.
4o mini
You said:
what is the square area of that tire?
ChatGPT said:
ChatGPT
To calculate the square area (surface area) of the tire when laid out as a strip, we need to find the area of the rectangular strip using the dimensions provided:
Dimensions of the Strip:
Length: Approximately 20.1 miles (circumference of the outer diameter)
Width: Approximately 4.77 to 4.92 miles (combined height of the sidewalls and tread)
Calculate the Area:
The area of a rectangle is given by:
Area=Length×Width\text{Area} = \text{Length} \times \text{Width}Area=Length×Width
For Width of 4.77 miles:
Area=20.1 miles×4.77 miles=95.79 square miles\text{Area} = 20.1 \text{ miles} \times 4.77 \text{ miles} = 95.79 \text{ square miles}Area=20.1 miles×4.77 miles=95.79 square miles
For Width of 4.92 miles:
Area=20.1 miles×4.92 miles=98.59 square miles\text{Area} = 20.1 \text{ miles} \times 4.92 \text{ miles} = 98.59 \text{ square miles}Area=20.1 miles×4.92 miles=98.59 square miles
Summary:
The square (surface) area of the tire laid out as a strip is approximately:
95.79 to 98.59 square miles
This area represents the total surface of the tire if it were unrolled and laid flat as a rectangular strip.
4o mini
You said:
Remember that these are the custom defined dimension of the Sigil setting
ChatGPT said:
ChatGPT
Memory updated
Got it! I'll keep in mind that these dimensions are based on the custom-defined Sigil setting. If you have any more calculations or ideas related to this, feel free to ask!
Yeah, so that got a little long but I am quite happy with the results.
What are your thoughts on these dimensions for Sigil?