tupper公式（塔珀自指公式）

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有关塔珀自指公式的简要笔记

简述

Tupper 的自引用公式是由 Jeff Tupper 定义的公式，当在平面中非常特定的位置以二维图形绘制时，可以直观地再现公式本身。

实际上，它再现了所有可能的 106 像素宽和 17 像素高的图像。

该公式是由以下定义的不等式：

\[ \frac{1}{2}<\left\lfloor\mathrm{mod}\left(\left\lfloor\frac{y}{17}\right\rfloor 2^{-17\lfloor x\rfloor-\mathrm{mod}(\lfloor y\rfloor,17))},2\right)\right\rfloor \] 对于不同的 \(k\) 值，可以求得不同的图像，而当 \(k\) 为

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将得到与自身表达式完全一样的图像，故被称作自引用（自指）公式。

自指公式图像

可以使用在线网站绘制图像：https://tuppers-formula.ovh/

Python实现

与原公式不相同，做了一些修正

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from typing import Literal

def tupper_self_referential_formula(k: int, width: int = 17, height: int = 106):
    """
    Tupper self referential formula函数

    Parameters:
        k - 用于迭代图像的k值
        width - 图像的宽
        height - 图像的高
    
    Returns:
        类型是numpy.ndarray矩阵，里面只有0和1
    """
    import numpy as np
    rst = np.zeros((width, height))
    def f(x: int, y: int) -> int:
        y += k
        a1 = 2 ** (width * x + y % width)
        a2 = (y // width) // a1
        return 1 if (a2 & 1) == 1 else 0
    
    for y in range(width):
        for x in range(height):
            rst[y, x] = f(x, y)
    
    return rst[:, ::-1]

def tupper_to_str(k: int, width: int = 17, height: int = 106, show_black: str = "■", show_white: str = "□") -> str:
    """
    生成Tupper self referential formula函数的字符画

    Parameters:
        k - 用于迭代图像的k值
        width - 图像的宽
        height - 图像的高
        show_black - 字符画中的黑色方块
        show_white - 字符画中的白色方块
    
    Returns:
        字符画字符串
    """
    mat = tupper_self_referential_formula(k, width, height)
    rst = ''
    for y in range(width):
        for x in range(height):
            if mat[y, x] == 1:
                rst += show_black
            else:
                rst += show_white
        rst += '\n'
    return rst

def tupper_to_img(k: int, width: int = 17, height: int = 106, filename: str = 'tupper.png', origin: Literal['upper', 'lower'] = 'upper'):
    """
    生成Tupper self referential formula函数的字符画

    Parameters:
        k - 用于迭代图像的k值
        width - 图像的宽
        height - 图像的高
        filename - 输出图像的文件名
        origin - 输出图像的方式(upper或者lower)
    
    Returns:
        类型是numpy.ndarray矩阵，里面只有0和1
    """
    import matplotlib.pyplot as plt
    rst = tupper_self_referential_formula(k, width, height)
    plt.figure(figsize=(15, 10))
    plt.imshow(rst, origin=origin)
    plt.savefig(filename)
    
    return rst


k = 92898203278702907929705938676672021500394791427205757369123489204565300324859717082409892641951206664564991991489354661871425872649524078000948199832659815275909285198829276929014694628110159824930931595166203271443269827449505707655085842563682060910813942504507936625555735585913273575050118552353192682955310220323463465408645422334101446471078933149287336241772448338428740302833855616421538520769267636119285948674549756604384946996184385407505456168240123319785800909933214695711828013483981731933773017336944656397583872267126767778549745087854794302808950100966582558761224454242018467578959766617176016660101690140279961968740323327369347164623746391335756442566959352876706364265509834319910419399748338894746638758652286771979896573695823608678008814861640308571256880794312652055957150464513950305355055495262375870102898500643010471425931450046440860841589302890250456138060738689526283389256801969190204127358098408264204643882520969704221896973544620102494391269663693407573658064279947688509910028257209987991480259150865283245150325813888942058

rst = tupper_to_str(k, 22, 160)
print(rst)
tupper_to_img(k, 22, 160)