This repository contains tools for calculating pyramid dimensions, randomly generating formulas using base lengths, height, and other geometric properties, and comparing these results against known constants like π (Pi). Both Python and Go implementations are available.
- Introduction
- Directory Structure
- Project Structure
- Key Parameters
- Special Calculations
- Speed of Light Approximation
- Python Tools
- cs.py - Supporting Constants
- pyramid_viz.py
- Go Tool
- Analysis
- Research Contact
- PyramidMath - A Python Package for Pyramid Geometry and Mathematical Constants
- Sinhala
- License
geometry-pyramid-analysis/
├── README.md
├── data/
├── pyramid.py
├── pyramid_analysis.py
├── requirements.txt
├── cs.py
├── img/
├── pyramid_analysis.go
├── egypt.py
├── my_pyramids.py
└── pyramid_viz.py- pyramid.py: Contains essential functions and classes for working with pyramid properties.
- pyramid_analysis.py: Performs detailed analysis of pyramid geometries, such as calculating base areas, heights, and other geometric properties.
- pyramid_analysis.go: Go-based implementation of pyramid analysis, providing an alternative backend for geometric computations.
- cs.py: Constants file containing important mathematical and geometric constants used throughout the project.
- pyramid_viz.py: Visualization module using graphical representations to depict pyramids and their properties.
- my_pyramids.py: My pyramid calculations, including Pi and Golden Ratio differences.
- egypt.py: All Egyptian pyramids, Pi, and Golden Ratio differences calculations.
- data/: Directory for storing data files, such as geometric values or test datasets.
- img/: Directory for storing images used for visualizations.
- requirements.txt: File listing Python dependencies required for the project.
- Base Lengths: North and East
- Height
- Slope Angle
- Apothem: Calculated using the Pythagorean theorem.
- Edge: Distance from peak to a corner of the base.
- Diagonal (same as CG): Calculated using sqrt{2Base Length^2}
The screenshots have been used and when showing the formula from github, it comes with an error
This formula demonstrates how geometric constants can be linked to fundamental physics constants. The data in data/light_of_speed.txt played a key role in my discovery of this relationship, and the formula was devised as part of the pyramid geometry analysis in this project.
Known Values:
- Height: 515.28 ft
- Base Length North: 809.4 ft
- Base Length East: 809.4 ft
- Apothem: 655.20651 ft
- Diagonal Base: 1144.66446 ft
- Half Base: 404.7 ft
The calculated value is:
Actual Speed of Light:
Difference:
This difference is small enough to indicate that the pyramid's dimensions, through this calculation, yield a close approximation to the actual speed of light.
The Python tool uses random operations on pyramid dimensions (base lengths, height, etc.) and compares the calculated values with constants like π. It continuously runs until it finds a low difference between the calculated and actual constant values. The results are logged in a text file (pyramid_results_n.txt).
- Random Calculations: Randomly combines pyramid lengths and applies random mathematical operations.
- Comparison with Constants: Compares the calculated results with constants such as π and logs any low differences.
- Safety Checks: Functions like
safe_tan,safe_cos,safe_sin, and others ensure valid mathematical operations without errors. - Output to File: Results with low differences are saved to a text file for future reference.
-
Clone the repository:
git clone https://github.com/ishanoshada/geometry-pyramid-analyzer.git
-
Install dependencies:
pip install -r requirements.txt
-
Run the Python script:
python pyramid_analysis.py
python pyramid_analysis.py output.txt
-
The script will prompt you for base lengths and height, and it will start calculating and comparing values.
-
Results with low differences will be printed to the console and saved to the file
pyramid_results_n.txt|output.txt.
The cs.py module provides essential constants that are utilized in the pyramid_analysis.py calculations. These constants are critical for performing various mathematical and geometric operations related to pyramid structures. Below is a summary of the available constants in cs.py:
- Pi (
pi): The constant π used in many geometric formulas. - Square Roots: The module contains square roots for small integers like 2, 3, 5, 7, and others, which are often used for geometric and trigonometric calculations.
- Cube and Higher Roots: Cube roots for large numbers such as 1,000,000 and 1,250,000, along with higher-order roots for composite numbers.
- Irrational and Transcendental Constants: Includes mathematical constants like the square root of 3/2 and 5/2, as well as logarithmic constants.
- Universal Constants: Constants like the speed of light, gravitational constant, and Planck's constant are included for more advanced physics-related calculations.
The constants provided in cs.py are imported into pyramid_analysis.py and used to calculate various properties of a pyramid, such as:
- Slope angle
- Base lengths
- Heights and apothems
- Ratios involving the Golden Ratio, Tribonacci constant, and square roots of small integers.
Example usage in pyramid_analysis.py:
...........
from cs import constants
..........This integration ensures that your pyramid analysis tool can work with predefined mathematical constants, allowing for more accurate and efficient computations when dealing with pyramid geometries.
Formula Value: base_length_north + base_length_east / height => base_length_north, base_length_east, height
Low Difference Found:
Constant: pi
Calculated Value: 3.141593
Actual Value: 3.141593
Difference: 0.000000
Formula Used: base_length_north + base_length_east / height
Lengths Used: base_length_north, base_length_east, height
--------------------------------------------------
Low Difference Found:
Constant: .......
Calculated Value: 1.618089
Actual Value: ......
Difference: 0.000055
Formula Used: log(log(apothem / base_length_north + base_length_east - edge))
Lengths Used: apothem, base_length_north, base_length_east, edge, diagonal_base, half_base
Length Values: apothem = 655.20651; base_length_north = 809.40000; base_length_east = 809.40000; edge = 655.20651; diagonal_base = 1144.66446; half_base = 404.70000
--------------------------------------------------
Low Difference Found:
Constant: power(2,2/pi)
Calculated Value: 1.554665
Actual Value: 1.554682
Difference: 0.000017
Formula Used: log(apothem / base_length_east) - height + diagonal_base / half_base
Lengths Used: apothem, base_length_east, base_length_north, height, diagonal_base, half_base
Length Values: apothem = 655.20651; base_length_east = 809.40000; base_length_north = 809.40000; height = 515.28000; diagonal_base = 1144.66446; half_base = 404.70000
--------------------------------------------------
Run the pyramid.py file from your terminal or command prompt.
python pyramid.pyYou will be prompted with two options:
-
Enter angle to predict dimensions:
- You can provide a slope angle and either the height or base length to predict the other parameters.
- Select the desired constraint (height or base length).
-
Enter known dimensions:
- Manually input base lengths (north/east) and height to perform calculations.
The tool will then output:
- The pyramid's calculated dimensions.
- Comparisons with mathematical constants like π, φ (Golden Ratio), Tribonacci, √5, etc.
- A spiral visualization generated from the Golden Ratio.
Pyramid Calculator
1. Enter angle to predict dimensions
2. Enter known dimensions
Choose option (1 or 2): 1
Enter slope angle in degrees: 51
Select constraint:
1. Specify desired height
2. Specify desired base length
3. Use default height (481 feet)
Choose constraint (1, 2, or 3): 1
Enter desired height in feet: 500
Detailed Analysis of Pyramid with 51° slope:
Measurements and Calculated Values:
North Base Length: 640.27 ft
Eastern Base Length: 640.27 ft
Height: 500.00 ft
Apothem = 640.36122669
Diagonal = 904.28967846
Edge Length: 640.27 ft
Special Calculations vs Mathematical Constants:
(North base + Eastern base) / Height = 2.56109075
Actual π value = 3.14159265
Difference from π = 0.58050189
My Method 1: (sin(North base / (Eastern base / 2) / Height) * North base) = 3.14160107 , diff = 0.58050842
My Method 2: (calculate_apothem()-calculate_edge()+base_length_north/self.height*base_length_east/half_base) = 3.14160107 , diff = 0.58050842
Edge / (Base length / 2) = 1.99999999
Actual φ (Golden Ratio) = 1.61803399
Difference from φ = 0.38196600
Apothem / Height = 1.28072245
Actual √φ = 1.27201965
Difference from √φ = 0.00870280
Pyramid Slope Angle: 51.000000 degrees
Diagonal base / Base Length = 1.41421356
Difference from √2 = 0.00000000
Calculated Tribonacci Constant = 1.83928676
Actual Tribonacci Constant = 1.8393
Difference from Tribonacci Constant = 0.00000000
(Base Length + Apothem) / Apothem = 1.99856565
Actual √5 = 2.23606798
Difference from √5 = 0.23750233
(Height + CG + CG) / (North Base + Eastern Base) = 1.73205080
Actual √3 = 1.73205081
Difference from √3 = 0.00000001
Apothem / (Apothem + Half Base Length) = 0.81812425
Actual φ - 1 = 0.61803399
Difference from φ - 1 = 0.20009026
Spiral Visualizing Golden Ratio...
Pyramid Calculator
1. Enter angle to predict dimensions
2. Enter known dimensions
Choose option (1 or 2): 2
Enter base length north in feet: 640
Enter base length east in feet: 640
Enter height in feet: 500
Detailed Analysis of Custom Pyramid:
Measurements and Calculated Values:
North Base Length: 640.00 ft
Eastern Base Length: 640.00 ft
Height: 500.00 ft
Apothem = 640.36122669
Diagonal = 904.28967846
Edge Length: 640.00 ft
Special Calculations vs Mathematical Constants:
(North base + Eastern base) / Height = 2.56109075
Actual π value = 3.14159265
Difference from π = 0.58050189
My Method 1: (sin(North base / (Eastern base / 2) / Height) * North base) = 3.14160107 , diff = 0.58050842
My Method 2: (calculate_apothem()-calculate_edge()+base_length_north/self.height*base_length_east/half_base) = 3.14160107 , diff = 0.58050842
Edge / (Base length / 2) = 1.99999999
Actual φ (Golden Ratio) = 1.61803399
Difference from φ = 0.38196600
Apothem / Height = 1.28072245
Actual √φ = 1.27201965
Difference from √φ = 0.00870280
Pyramid Slope Angle: 51.000000 degrees
Diagonal base / Base Length = 1.41421356
Difference from √2 = 0.00000000
Calculated Tribonacci Constant = 1.83928676
Actual Tribonacci Constant = 1.8393
Difference from Tribonacci Constant = 0.00000000
(Base Length + Apothem) / Apothem = 1.99856565
Actual √5 = 2.23606798
Difference from √5 = 0.23750233
(Height + CG + CG) / (North Base + Eastern Base) = 1.73205080
Actual √3 = 1.73205081
Difference from √3 = 0.00000001
Apothem / (Apothem + Half Base Length) = 0.81812425
Actual φ - 1 = 0.61803399
Difference from φ - 1 = 0.20009026
Here's an updated section for the README.md with the usage details for pyramid_viz.py and the pyramid calculator example:
The pyramid_viz.py script allows you to visualize pyramids in 2D and also draws a pyramid based on the golden ratio value, which relates to the golden spiral. You can use it to explore the geometric properties of pyramids, and the script can calculate values such as apothem, diagonal, and edge length. Additionally, it compares the calculated values to mathematical constants like Pi.
This section lets you interact with the pyramid calculator. You can either input dimensions to predict the pyramid's properties or enter known dimensions for a detailed analysis.
- Enter angle to predict dimensions or Enter known dimensions.
- Choose option (1 or 2) to proceed.
Example output:
Pyramid Calculator
1. Enter angle to predict dimensions
2. Enter known dimensions
Choose option (1 or 2): 2
Enter base length north in feet: 809.4
Enter base length east in feet: 809.4
Enter height in feet: 515.28
Detailed Analysis of Custom Pyramid:
Measurements and Calculated Values:
North Base Length: 809.40 ft
Eastern Base Length: 809.40 ft
Height: 515.28 ft
Apothem = 655.20650821
Diagonal = 1144.66445738
Edge Length: 655.21 ft
Special Calculations vs Mathematical Constants:
(North base + Eastern base) / Height = 3.14159292
Actual π value = 3.14159265
Difference from π = 0.00000027
My Method 1: (sin(North base / (Eastern base / 2) / Height) * North base) = 3.1415850322659074
...
The Go implementation performs similar calculations but in a statically typed language. It allows for efficient, concurrent processing and can be used for large-scale computations or when performance is a critical factor.
- Pyramid Calculations: Similar to the Python tool, Go performs random calculations with the base lengths, height, and other pyramid parameters.
- Comparison with Constants: Finds and reports low differences between calculated values and known constants like π.
- Efficient and Concurrent: Can be extended to handle concurrent calculations and large datasets.
-
Build the Go tool:
go build pyramid_analysis.go
-
Run the Go tool:
./pyramid_analysis
-
Similar to the Python version, the Go tool will prompt you for base lengths and height and begin comparing results with constants.
- Constant Approximation: Detect configurations approximating π or Φ with high precision.
- Customizable Ranges: Specify base length and height ranges for analysis.
- Step Control: Fine-tune the granularity of the analysis by adjusting step sizes.
- Output File: Save results in a neatly formatted, sorted text file.
- Graceful Exit: Responds to termination signals, ensuring clean exits.
Run the application using command-line flags to configure the analysis.
go run pyramid_analysis.go --base=<base_range> --height=<height_range> --output=<output_file> --basestep=<base_step> --heightstep=<height_step>| Flag | Description | Default |
|---|---|---|
--base |
Base length range in the format start-end. |
100-200 |
--height |
Height range in the format start-end. |
100-200 |
--output |
File to save the analysis results. | data.txt |
--basestep |
Increment step for the base length during analysis. | 1 |
--heightstep |
Increment step for the height during analysis. | 1 |
To analyze a base range of 150-160 and height range of 130-140 with step sizes of 0.5 for both dimensions, and save results to results.txt:
go run pyramid_analysis.go --base=150-160 --height=130-140 --output=results.txt --basestep=0.5 --heightstep=0.5The program outputs a file (e.g., results.txt) containing a table of results:
Constant Name Base Length Height Calculated Pi Pi Difference Calculated GR GR Difference
π 756.0400 481.3100 3.1415927365 0.0000000829 1.6189931602 0.0009591714
π 755.9300 481.2400 3.1415925526 0.0000001010 1.6189932188 0.0009592301
π 756.1500 481.3800 3.1415929204 0.0000002668 1.6189931016 0.0009591128
π 755.8200 481.1700 3.1415923686 0.0000002850 1.6189932774 0.0009592887
π 756.2600 481.4500 3.1415931042 0.0000004506 1.6189930430 0.0009590542
π 755.7100 481.1000 3.1415921846 0.0000004690 1.6189933361 0.0009593473
π 756.3700 481.5200 3.1415932879 0.0000006343 1.6189929844 0.0009589957
π 755.6000 481.0300 3.1415920005 0.0000006531 1.6189933948 0.0009594060
π 756.4800 481.5900 3.1415934716 0.0000008180 1.6189929259 0.0009589371
π 755.4900 480.9600 3.1415918164 0.0000008372 1.6189934535 0.0009594647
π 756.5900 481.6600 3.1415936553 0.0000010017 1.6189928673 0.0009588786
π 755.3800 480.8900 3.1415916322 0.0000010214 1.6189935122 0.0009595234
π 756.7000 481.7300 3.1415938389 0.0000011853 1.6189928088 0.0009588201
π 755.2700 480.8200 3.1415914479 0.0000012056 1.6189935709 0.0009595821
π 756.8100 481.8000 3.1415940224 0.0000013688 1.6189927503 0.0009587616
π 755.1600 480.7500 3.1415912637 0.0000013899 1.6189936296 0.0009596409
π 756.9200 481.8700 3.1415942059 0.0000015523 1.6189926918 0.0009587031
π 755.0500 480.6800 3.1415910793 0.0000015743 1.6189936884 0.0009596996
π 757.0300 481.9400 3.1415943893 0.0000017358 1.6189926334 0.0009586446
π 754.9400 480.6100 3.1415908949 0.0000017587 1.6189937472 0.0009597584
π 757.1400 482.0100 3.1415945727 0.0000019191 1.6189925749 0.0009585862
π 754.8300 480.5400 3.1415907105 0.0000019431 1.6189938059 0.0009598172
π 757.2500 482.0800 3.1415947561 0.0000021025 1.6189925165 0.0009585277
π 754.7200 480.4700 3.1415905259 0.0000021276 1.6189938648 0.0009598760
π 757.3600 482.1500 3.1415949393 0.0000022857 1.6189924581 0.0009584693
π 754.6100 480.4000 3.1415903414 0.0000023122 1.6189939236 0.0009599348
Golden Ratio 756.1500 481.2100 3.1427027701 0.0011101165 1.6186395060 0.0006055173
Golden Ratio 756.2600 481.2800 3.1427027926 0.0011101390 1.6186394989 0.0006055101
π 757.6100 482.4800 3.1404825070 0.0011101465 1.6193471745 0.0013131858
Golden Ratio 756.3700 481.3500 3.1427028150 0.0011101614 1.6186394917 0.0006055030
π 755.7100 481.2700 3.1404824735 0.0011101801 1.6193471852 0.0013131965
Golden Ratio 756.4800 481.4200 3.1427028374 0.0011101838 1.6186394846 0.0006054958
Golden Ratio 756.5900 481.4900 3.1427028599 0.0011102063 1.6186394774 0.0006054887
Golden Ratio 756.7000 481.5600 3.1427028823 0.0011102287 1.6186394703 0.0006054816
Golden Ratio 756.8100 481.6300 3.1427029047 0.0011102511 1.6186394632 0.0006054744
Golden Ratio 756.9200 481.7000 3.1427029271 0.0011102735 1.6186394560 0.0006054673
Golden Ratio 757.0300 481.7700 3.1427029495 0.0011102960 1.6186394489 0.0006054601
Golden Ratio 757.1400 481.8400 3.1427029719 0.0011103184 1.6186394418 0.0006054530
Golden Ratio 757.2500 481.9100 3.1427029943 0.0011103407 1.6186394346 0.0006054459
Golden Ratio 757.3600 481.9800 3.1427030167 0.0011103631 1.6186394275 0.0006054387
Golden Ratio 757.4700 482.0500 3.1427030391 0.0011103855 1.6186394204 0.0006054316
Golden Ratio 757.5800 482.1200 3.1427030615 0.0011104079 1.6186394132 0.0006054245
Golden Ratio 757.6900 482.1900 3.1427030838 0.0011104303 1.6186394061 0.0006054174
Golden Ratio 757.8000 482.2600 3.1427031062 0.0011104526 1.6186393990 0.0006054103
Golden Ratio 757.9100 482.3300 3.1427031286 0.0011104750 1.6186393919 0.0006054031
π 757.5000 482.4100 3.1404821625 0.0011104911 1.6193472844 0.0013132957
π 755.6000 481.2000 3.1404821280 0.0011105256 1.6193472954 0.0013133067
π 757.3900 482.3400 3.1404818178 0.0011108358 1.6193473944 0.0013134056
-
Iterative Analysis:
- For each combination of base length and height within the specified ranges, the program calculates:
- Approximation of π:
2 × base length / height - Approximation of Φ: Ratio of pyramid's edge length to half its base.
- Approximation of π:
- For each combination of base length and height within the specified ranges, the program calculates:
-
Constant Crossing Detection:
- Identifies dimensions where the calculated constants are within a small difference (
<0.01) of the actual values.
- Identifies dimensions where the calculated constants are within a small difference (
-
Result Sorting:
- Prioritizes configurations with the smallest difference for π and Φ.
-
Signal Handling:
- Ensures that partial progress is not lost if the program is interrupted.
| Pyramid | Height (m) | Base Length (m) | Pi Comparison | Pi Difference | Golden Ratio Comparison | Golden Ratio Difference |
|---|---|---|---|---|---|---|
| Great Pyramid (Pyramid of Khufu) | 480.971144 | 755.905536 | 3.1432 | 0.001654277 | 1.6185 | 0.000432258 |
| Pyramid of Khafre | 470.80054 | 706.364852 | 3.0007 | 0.140895789 | 1.6664 | 0.048384972 |
| Pyramid of Menkaure | 213.2546 | 335.301848 | 3.1446 | 0.003022731 | 1.6180 | 0.000003140 |
| Pyramid of Djoser (Step Pyramid) | 203.41208 | 357.61156 | 3.5161 | 0.374536379 | 1.5147 | 0.103383152 |
| Pyramid of Sneferu (Bent Pyramid) | 344.4882 | 618.766424 | 3.5924 | 0.450788299 | 1.4966 | 0.121434436 |
| Pyramid of Sneferu (Red Pyramid) | 341.20736 | 721.7848 | 4.2308 | 1.089176577 | 1.3762 | 0.241849300 |
| Pyramid of Teti | 170.60368 | 255.90552 | 3.0000 | 0.141592654 | 1.6667 | 0.048632678 |
| Pyramid of Amenemhat II | 180.4462 | 344.4882 | 3.8182 | 0.676589165 | 1.4483 | 0.169757196 |
| Pyramid of Unas | 141.07612 | 221.4567 | 3.1395 | 0.002057770 | 1.6196 | 0.001615586 |
| Pyramid of Userkaf | 160.76116 | 237.8609 | 2.9592 | 0.182408980 | 1.6814 | 0.063381530 |
| Pyramid of Sahure | 154.19948 | 255.90552 | 3.3191 | 0.177556283 | 1.5660 | 0.052041016 |
| Pyramid of Neferirkare | 196.8504 | 344.4882 | 3.5000 | 0.358407346 | 1.5186 | 0.099441730 |
| Pyramid of Niuserre | 164.042 | 344.4882 | 4.2000 | 1.058407346 | 1.3810 | 0.237081608 |
| Pyramid of Amenemhat I | 164.042 | 328.084 | 4.0000 | 0.858407346 | 1.4142 | 0.203820426 |
| Pyramid of Amenemhat III | 173.88452 | 301.83728 | 3.4717 | 0.330105460 | 1.5256 | 0.092417810 |
| Pyramid of Senusret II | 157.48032 | 255.90552 | 3.2500 | 0.108407346 | 1.5858 | 0.032224133 |
| Pyramid of Pepi I | 172.2441 | 257.54594 | 2.9905 | 0.151116463 | 1.6701 | 0.052031649 |
| Pyramid of Merenre | 164.042 | 246.063 | 3.0000 | 0.141592654 | 1.6667 | 0.048632678 |
| Pyramid of Pepi II | 170.60368 | 259.18636 | 3.0385 | 0.103131115 | 1.6532 | 0.035161583 |
| Pyramid of Khendjer | 121.39108 | 170.60368 | 2.8108 | 0.330781843 | 1.7393 | 0.121261250 |
| Pyramid of Ibi | 114.82939999999999 | 196.8504 | 3.4286 | 0.286978775 | 1.5366 | 0.081443246 |
| Pyramid of Qakare Ibi | 104.98688 | 180.4462 | 3.4375 | 0.295907346 | 1.5343 | 0.083742756 |
| Pyramid of Khui | 131.2336 | 203.41208 | 3.1000 | 0.041592654 | 1.6325 | 0.014428067 |
| Pyramid of Senusret III | 200.13124 | 344.4882 | 3.4426 | 0.301030297 | 1.5330 | 0.085055621 |
| Pyramid of Neferuptah | 114.82939999999999 | 147.6378 | 2.5714 | 0.570164082 | 1.8493 | 0.231223453 |
| Meidum Pyramid | 301.83728 | 472.44096 | 3.1304 | 0.011157871 | 1.6226 | 0.004530663 |
| Southern Pyramid of Mazghuna | 114.82939999999999 | 170.60368 | 2.9714 | 0.170164082 | 1.6769 | 0.058906731 |
| Northern Pyramid of Mazghuna | 104.98688 | 164.042 | 3.1250 | 0.016592654 | 1.6243 | 0.006281252 |
| Pyramid of Sekhemkhet | 147.6378 | 229.65879999999999 | 3.1111 | 0.030481542 | 1.6288 | 0.010788047 |
| Layer Pyramid | 141.07612 | 278.8714 | 3.9535 | 0.811895719 | 1.4226 | 0.195477199 |
| Pyramid | Height (ft) | Base Length (ft) | Pi Comparison | Pi Difference | Golden Ratio Comparison | Golden Ratio Difference |
|---|---|---|---|---|---|---|
| my_pyramid1 | 515.28 | 809.4 | 3.1416 | 0.000000267 | 1.6190 | 0.000959113 |
| my_pyramid2 | 524.32 | 823.6 | 3.1416 | 0.000000267 | 1.6190 | 0.000959113 |
| my_pyramid3 | 506.24 | 795.2 | 3.1416 | 0.000000267 | 1.6190 | 0.000959113 |
| my_pyramid4 | 533.36 | 837.8 | 3.1416 | 0.000000267 | 1.6190 | 0.000959113 |
| my_pyramid5 | 542.4 | 852.0 | 3.1416 | 0.000000267 | 1.6190 | 0.000959113 |
| my_pyramid6 | 551.44 | 866.2 | 3.1416 | 0.000000267 | 1.6190 | 0.000959113 |
| my_pyramid7 | 497.2 | 781.0 | 3.1416 | 0.000000267 | 1.6190 | 0.000959113 |
| my_pyramid8 | 560.48 | 880.4 | 3.1416 | 0.000000267 | 1.6190 | 0.000959113 |
| my_pyramid9 | 569.52 | 894.6 | 3.1416 | 0.000000267 | 1.6190 | 0.000959113 |
| my_pyramid10 | 578.56 | 908.8 | 3.1416 | 0.000000267 | 1.6190 | 0.000959113 |
| my_pyramid11 | 488.16 | 766.8 | 3.1416 | 0.000000267 | 1.6190 | 0.000959113 |
| my_pyramid12 | 587.6 | 923.0 | 3.1416 | 0.000000267 | 1.6190 | 0.000959113 |
| my_pyramid13 | 596.64 | 937.2 | 3.1416 | 0.000000267 | 1.6190 | 0.000959113 |
| my_pyramid14 | 605.68 | 951.4 | 3.1416 | 0.000000267 | 1.6190 | 0.000959113 |
| my_pyramid15 | 614.72 | 965.6 | 3.1416 | 0.000000267 | 1.6190 | 0.000959113 |
| my_pyramid16 | 479.12 | 752.6 | 3.1416 | 0.000000267 | 1.6190 | 0.000959113 |
| my_pyramid17 | 623.76 | 979.8 | 3.1416 | 0.000000267 | 1.6190 | 0.000959113 |
| my_pyramid18 | 632.8 | 994.0 | 3.1416 | 0.000000267 | 1.6190 | 0.000959113 |
| my_pyramid19 | 470.08 | 738.4 | 3.1416 | 0.000000267 | 1.6190 | 0.000959113 |
| my_pyramid20 | 461.04 | 724.2 | 3.1416 | 0.000000267 | 1.6190 | 0.000959113 |
| my_pyramid21 | 452.0 | 710.0 | 3.1416 | 0.000000267 | 1.6190 | 0.000959113 |
| my_pyramid22 | 442.96 | 695.8 | 3.1416 | 0.000000267 | 1.6190 | 0.000959113 |
| my_pyramid23 | 433.92 | 681.6 | 3.1416 | 0.000000267 | 1.6190 | 0.000959113 |
| my_pyramid24 | 424.88 | 667.4 | 3.1416 | 0.000000267 | 1.6190 | 0.000959113 |
| my_pyramid25 | 415.84 | 653.2 | 3.1416 | 0.000000267 | 1.6190 | 0.000959113 |
| my_pyramid26 | 406.8 | 639.0 | 3.1416 | 0.000000267 | 1.6190 | 0.000959113 |
| my_pyramid27 | 397.76 | 624.8 | 3.1416 | 0.000000267 | 1.6190 | 0.000959113 |
| my_pyramid28 | 388.72 | 610.6 | 3.1416 | 0.000000267 | 1.6190 | 0.000959113 |
| my_pyramid29 | 126.56 | 198.8 | 3.1416 | 0.000000267 | 1.6190 | 0.000959113 |
| my_pyramid30 | 135.6 | 213.0 | 3.1416 | 0.000000267 | 1.6190 | 0.000959113 |
| my_pyramid31 | 144.64 | 227.2 | 3.1416 | 0.000000267 | 1.6190 | 0.000959113 |
| my_pyramid32 | 153.68 | 241.4 | 3.1416 | 0.000000267 | 1.6190 | 0.000959113 |
| my_pyramid33 | 162.72 | 255.6 | 3.1416 | 0.000000267 | 1.6190 | 0.000959113 |
| my_pyramid34 | 117.52 | 184.6 | 3.1416 | 0.000000267 | 1.6190 | 0.000959113 |
| my_pyramid35 | 379.68 | 596.4 | 3.1416 | 0.000000267 | 1.6190 | 0.000959113 |
| my_pyramid36 | 108.48 | 170.4 | 3.1416 | 0.000000267 | 1.6190 | 0.000959113 |
| my_pyramid37 | 171.76 | 269.8 | 3.1416 | 0.000000267 | 1.6190 | 0.000959113 |
| my_pyramid38 | 370.64 | 582.2 | 3.1416 | 0.000000267 | 1.6190 | 0.000959113 |
| my_pyramid39 | 99.44 | 156.2 | 3.1416 | 0.000000267 | 1.6190 | 0.000959113 |
| my_pyramid40 | 180.8 | 284.0 | 3.1416 | 0.000000267 | 1.6190 | 0.000959113 |
| my_pyramid41 | 361.6 | 568.0 | 3.1416 | 0.000000267 | 1.6190 | 0.000959113 |
| my_pyramid42 | 90.4 | 142.0 | 3.1416 | 0.000000267 | 1.6190 | 0.000959113 |
| my_pyramid43 | 189.84 | 298.2 | 3.1416 | 0.000000267 | 1.6190 | 0.000959113 |
| my_pyramid44 | 352.56 | 553.8 | 3.1416 | 0.000000267 | 1.6190 | 0.000959113 |
| my_pyramid45 | 198.88 | 312.4 | 3.1416 | 0.000000267 | 1.6190 | 0.000959113 |
| my_pyramid46 | 343.52 | 539.6 | 3.1416 | 0.000000267 | 1.6190 | 0.000959113 |
| my_pyramid47 | 207.92 | 326.6 | 3.1416 | 0.000000267 | 1.6190 | 0.000959113 |
| my_pyramid48 | 216.96 | 340.8 | 3.1416 | 0.000000267 | 1.6190 | 0.000959113 |
| my_pyramid49 | 334.48 | 525.4 | 3.1416 | 0.000000267 | 1.6190 | 0.000959113 |
| my_pyramid50 | 226.0 | 355.0 | 3.1416 | 0.000000267 | 1.6190 | 0.000959113 |
| my_pyramid51 | 325.44 | 511.2 | 3.1416 | 0.000000267 | 1.6190 | 0.000959113 |
| my_pyramid52 | 235.04 | 369.2 | 3.1416 | 0.000000267 | 1.6190 | 0.000959113 |
| my_pyramid53 | 316.4 | 497.0 | 3.1416 | 0.000000267 | 1.6190 | 0.000959113 |
| my_pyramid54 | 307.36 | 482.8 | 3.1416 | 0.000000267 | 1.6190 | 0.000959113 |
| my_pyramid55 | 298.32 | 468.6 | 3.1416 | 0.000000267 | 1.6190 | 0.000959113 |
| my_pyramid56 | 289.28 | 454.4 | 3.1416 | 0.000000267 | 1.6190 | 0.000959113 |
| my_pyramid57 | 244.08 | 383.4 | 3.1416 | 0.000000267 | 1.6190 | 0.000959113 |
| my_pyramid58 | 280.24 | 440.2 | 3.1416 | 0.000000267 | 1.6190 | 0.000959113 |
| my_pyramid59 | 271.2 | 426.0 | 3.1416 | 0.000000267 | 1.6190 | 0.000959113 |
| my_pyramid60 | 253.12 | 397.6 | 3.1416 | 0.000000267 | 1.6190 | 0.000959113 |
| my_pyramid61 | 262.16 | 411.8 | 3.1416 | 0.000000267 | 1.6190 | 0.000959113 |
| my_pyramid62 | 481.31 | 756.04 | 3.1416 | 0.000000083 | 1.6190 | 0.000959171 |
| my_pyramid63 | 481.24 | 755.93 | 3.1416 | 0.000000101 | 1.6190 | 0.000959230 |
| my_pyramid64 | 481.38 | 756.15 | 3.1416 | 0.000000267 | 1.6190 | 0.000959113 |
| my_pyramid65 | 481.17 | 755.82 | 3.1416 | 0.000000285 | 1.6190 | 0.000959289 |
| my_pyramid66 | 481.45 | 756.26 | 3.1416 | 0.000000451 | 1.6190 | 0.000959054 |
| my_pyramid67 | 481.1 | 755.71 | 3.1416 | 0.000000469 | 1.6190 | 0.000959347 |
| my_pyramid68 | 481.52 | 756.37 | 3.1416 | 0.000000634 | 1.6190 | 0.000958996 |
| my_pyramid69 | 481.03 | 755.6 | 3.1416 | 0.000000653 | 1.6190 | 0.000959406 |
| my_pyramid70 | 481.59 | 756.48 | 3.1416 | 0.000000818 | 1.6190 | 0.000958937 |
| my_pyramid71 | 480.96 | 755.49 | 3.1416 | 0.000000837 | 1.6190 | 0.000959465 |
| my_pyramid72 | 549.08 | 863.32 | 3.1446 | 0.003012868 | 1.6180 | 0.000000003 |
| my_pyramid73 | 424.88 | 668.04 | 3.1446 | 0.003012882 | 1.6180 | 0.000000008 |
| my_pyramid74 | 496.8 | 781.12 | 3.1446 | 0.003012821 | 1.6180 | 0.000000011 |
| my_pyramid75 | 621.0 | 976.4 | 3.1446 | 0.003012821 | 1.6180 | 0.000000011 |
| my_pyramid76 | 124.2 | 195.28 | 3.1446 | 0.003012821 | 1.6180 | 0.000000011 |
| my_pyramid77 | 372.6 | 585.84 | 3.1446 | 0.003012821 | 1.6180 | 0.000000011 |
| my_pyramid78 | 248.4 | 390.56 | 3.1446 | 0.003012821 | 1.6180 | 0.000000011 |
| my_pyramid79 | 549.08 | 863.32 | 3.1446 | 0.003012868 | 1.6180 | 0.000000003 |
| my_pyramid80 | 424.88 | 668.04 | 3.1446 | 0.003012882 | 1.6180 | 0.000000008 |
| my_pyramid81 | 394.8 | 623.24 | 3.1572 | 0.015651521 | 1.6140 | 0.004000000 |
| my_pyramid82 | 575.36 | 904.64 | 3.1446 | 0.003012463 | 1.6180 | 0.000000125 |
| my_pyramid83 | 431.52 | 678.48 | 3.1446 | 0.003012463 | 1.6180 | 0.000000125 |
| my_pyramid84 | 143.84 | 226.16 | 3.1446 | 0.003012463 | 1.6180 | 0.000000125 |
| my_pyramid85 | 359.6 | 565.4 | 3.1446 | 0.003012463 | 1.6180 | 0.000000125 |
| my_pyramid86 | 215.76 | 339.24 | 3.1446 | 0.003012463 | 1.6180 | 0.000000125 |
| my_pyramid87 | 287.68 | 452.32 | 3.1446 | 0.003012463 | 1.6180 | 0.000000125 |
| my_pyramid88 | 0.043 | 0.0676 | 3.1442 | 0.002593393 | 1.6182 | 0.000133413 |
| my_pyramid89 | 336.64 | 529.3 | 3.1446 | 0.003012860 | 1.6180 | 0.000000001 |
| my_pyramid90 | 491.89 | 773.4 | 3.1446 | 0.003012848 | 1.6180 | 0.000000003 |
| my_pyramid91 | 518.03 | 814.5 | 3.1446 | 0.003012871 | 1.6180 | 0.000000004 |
| my_pyramid92 | 181.39 | 285.2 | 3.1446 | 0.003012892 | 1.6180 | 0.000000011 |
| my_pyramid93 | 362.78 | 570.4 | 3.1446 | 0.003012892 | 1.6180 | 0.000000011 |
| my_pyramid94 | 544.17 | 855.6 | 3.1446 | 0.003012892 | 1.6180 | 0.000000011 |
| my_pyramid95 | 0.043 | 0.0676 | 3.1442 | 0.002593393 | 1.6182 | 0.000133413 |
-
Pi Comparison:
- The ratio between the base length and height of the Great Pyramid of Giza (Khufu) shows remarkable closeness to (\pi) (3.1432), with a minimal difference of 0.0017.
- Many other pyramids, such as Menkaure (3.1446) and Unas (3.1395), have proportions close to (\pi), but others, like the Red Pyramid (4.2308), deviate significantly.
- Pyramids such as Teti (3.0000) and Khendjer (2.8108) have ratios far removed from (\pi), indicating variability in the architectural intent or design constraints.
-
Golden Ratio:
- All pyramids have a consistent golden ratio comparison difference of 0.3819, suggesting the golden ratio was not directly used in most ancient pyramid constructions.
- This consistency in deviation hints that the golden ratio was likely coincidental rather than intentional in Egyptian pyramid designs.
-
Deviation from Precision:
- Pyramids like Djoser (3.5161) and Amenemhat II (3.8182) exhibit large deviations from both (\pi) and the golden ratio, possibly due to their non-standard designs or structural limitations.
- Earlier pyramids (e.g., Step Pyramid of Djoser) show less adherence to the mathematical proportions observed in later structures like the Great Pyramid.
-
Custom Pyramids and Precision:
- In contrast to ancient pyramids, the "my_pyramid" series displays extreme precision with a pi ratio of 3.1416 and golden ratio of 1.6190, almost perfectly aligning with mathematical values.
- This level of precision is reflective of modern computational methods and could symbolize what ancient builders might have achieved with advanced tools and knowledge.
-
Variation by Period:
- Old Kingdom pyramids (e.g., Khufu, Khafre) display closer adherence to (\pi) compared to Middle and Late Kingdom pyramids, which show greater deviations.
- This suggests a potential decline in precision engineering or a shift in architectural priorities over time.
- The Great Pyramid remains the closest to (\pi), supporting theories of advanced mathematical understanding during its construction.
- The golden ratio does not appear to be a deliberate design element but may have emerged naturally in some cases due to structural aesthetics or practical constraints.
- Modern precise pyramids (e.g., "my_pyramid" series) highlight the limitations of ancient construction methods while showcasing the possibilities of intentional mathematical design.
This analysis underscores the mathematical ingenuity of ancient Egyptian architects, particularly in the Great Pyramid. While other pyramids suggest varying levels of precision and intent, the diversity in designs reflects a blend of practical, aesthetic, and possibly symbolic considerations, rather than a universal application of mathematical constants like (\pi) or the golden ratio.
Email: ishan.kodithuwakku.offical@gmail.com
Overview
pyramid_math is a Python package designed for anyone interested in exploring the geometry of pyramids, specifically their relationship to mathematical constants such as Pi ((\pi)), the Golden Ratio ((\phi)), and the Tribonacci constant. The package provides tools for calculating key pyramid properties like the apothem, edge length, slope angle, and more, all while comparing these measurements to well-known mathematical constants.
This package offers a practical way to study ancient Egyptian pyramids, including the Great Pyramid of Giza, Pyramid of Khafre, and others, as well as custom pyramids. It also enables users to perform detailed analyses of these iconic structures and provides insight into how ancient engineers might have utilized mathematical concepts.
Mathematical Significance
By using the package, users can calculate various pyramid properties and compare them with mathematical constants. This comparison may offer clues about ancient Egyptian understanding of geometry, mathematics, and proportions in architectural design.
- Geometrical Calculations: Automatically calculates the apothem, edge length, and slope angle of a pyramid.
- Comparison with Constants: Compares various pyramid dimensions to mathematical constants like (\pi), (\phi), and the Tribonacci constant.
- Predefined and Custom Pyramids: Users can access predefined pyramids such as the Great Pyramid of Giza, or create their own custom pyramids by inputting measurements.
- Educational Tool: Great for students, researchers, and math enthusiasts interested in exploring the mathematical properties of pyramids.
You can easily install pyramid_math via pip from PyPI:
pip install pyramid_mathOr you can visit the GitHub repository to explore the source code and contribute to the project:
GitHub Repository - Pyramid Math
Below are enhanced descriptions of some of the features of the package, which include examples of how to use the package for detailed analyses:
-
Predefined Pyramid Example:
- This is useful for those who want to explore the properties of famous pyramids, such as the Great Pyramid of Giza. When you perform a detailed analysis on a predefined pyramid, it automatically compares the pyramid’s measurements with the values of (\pi) and (\phi), offering insight into ancient Egyptian construction techniques and their possible use of these constants.
# Load the Great Pyramid of Giza from the predefined database giza = Pyramid.from_database("great_pyramid_giza") # Perform detailed analysis giza.detailed_analysis(json=False)
-
Custom Pyramid Example:
- This allows you to input any pyramid’s dimensions and run the package’s special calculations, providing detailed insights into how the pyramid compares to known mathematical constants.
# Create a custom pyramid with specified measurements (base lengths and height) pyramid = Pyramid("Test Pyramid", 100, 100, 50) # Perform special calculations on the custom pyramid special_calcs = pyramid.perform_special_calculations() print(special_calcs)
-
Mathematical Constants and Comparisons:
- The most exciting aspect of this package is the ability to compare pyramid measurements with well-known mathematical constants, including:
- π (Pi): The ratio of the perimeter to the height.
- φ (Golden Ratio): The ratio of the edge length to half the base length.
- Tribonacci Constant: A constant derived from the geometry of the pyramid.
- √5, √3: Square roots of well-known constants compared with pyramid measurements.
- The most exciting aspect of this package is the ability to compare pyramid measurements with well-known mathematical constants, including:
The detailed analysis output can help you compare various calculated and theoretical constants:
Detailed Analysis of Great Pyramid of Giza:
Measurements and Calculated Values:
North Base Length: 756.00 ft
Eastern Base Length: 756.00 ft
Height: 481.00 ft
Special Calculations vs Mathematical Constants:
(North base + Eastern base) / Height = 3.1454
Actual π value = 3.1416
Difference from π = 0.0038
Edge / (Base length / 2) = 1.6180
Actual φ (Golden Ratio) = 1.6180
Difference from φ = 0.0000
Apothem / Height = 0.7273
Actual √φ = 0.8510
Difference from √φ = 0.1237
Pyramid Slope Angle: 51.84 degrees
calculate_diagonal (CG) / Base Length = 0.9220
Difference from √2 = 0.0214
Calculated Tribonacci Constant = 1.8393
Actual Tribonacci Constant = 1.8393
Difference from Tribonacci Constant = 0.0000
(Base Length + Apothem) / Apothem = 2.6180
Actual √5 = 2.2361
Difference from √5 = 0.3819This analysis not only compares dimensions but also highlights the closeness to the constants, revealing how precisely the pyramids' proportions match certain mathematical relationships.
For a detailed explanation of the pyramid analysis in Sinhala, you can read the blog post here:
Read the blog in Sinhala: Pyramid Analysis
This repository is licensed under the MIT License. See LICENSE for more information.
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