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法界缘起是什么

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分形几何给出了答案

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一九八零年

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一个数学家往电脑里输入了一条公式

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z等于z的平方加c

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五个符号

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一行代码

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他按下回车

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屏幕上出现了一个形状

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他把那个形状放大

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放大一倍

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边界上浮现出新的细节

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放大十倍

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那些细节里藏着更小的相同形状

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放大一百倍 一千倍 一百万倍

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永远看不到尽头

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每一层放大里

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都在重复整体的结构

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这个数学家叫曼德勃罗

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这个形状后来被命名为曼德勃罗集

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它用人类已知最简单的规则

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生成了一个无限复杂的宇宙

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曼德勃罗不是纯粹的学院派

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他出生在波兰华沙的一个犹太家庭

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一九三六年随家人移居法国

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在那里经历了整个二战

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从小跟着做数学家的叔叔自学

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他后来进了IBM研究院

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在那个年代

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他是少数能用电脑做可视化的数学家

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别人在纸上算公式

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他在屏幕上看图形

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这个习惯改变了数学的方向

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因为有些东西

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你算一辈子也算不出它的形状

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但你画出来的那一刻

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答案就跳到你面前了

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哈喽

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大家好

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我是王利杰

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一千三百年前

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华严宗三祖法藏站在武则天面前

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试图解释一个他用语言说不清楚的东西

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他说

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一粒微尘里

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包含着整个法界

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不是反映

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不是代表

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是包含

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每一个局部

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就是全体

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这不是诗

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这是华严宗最核心的教义

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叫法界缘起

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先解释一下这四个字

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法界

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是佛学里说的一切存在的总体

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你可以简单理解为整个宇宙和宇宙里发生的一切

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缘起

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是说万事万物都依赖条件而生起

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没有任何东西能独立存在

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法界缘起把这两个概念合在一起

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说的是

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不光万物互相依赖

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而且每一个局部的万物互相依赖的结构

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跟整个宇宙的万物互相依赖的结构

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是完全一样的

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这不是普通的缘起

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这是缘起的缘起

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是结构本身在每一层尺度上的自我复制

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上一期我们聊了因陀罗网

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那张巨网上的每颗宝珠映照着所有其他宝珠

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物理学用全息原理告诉我们

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三维空间的信息确实可以编码在二维边界上

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局部可以重建全体的信息

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但我今天想说的是

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映照和包含不是一回事

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映照是影子

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包含是本体

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因陀罗网说的是投影关系

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法界缘起说的更狠 每一个局部的存在本身

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就已经是整体

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这个说法听起来像是纯粹的玄学

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直到二十世纪下半叶

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数学家发现了分形几何

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在曼德勃罗之前

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有一个更早的例子

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一九零四年

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瑞典数学家科赫画了一条雪花曲线

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方法很简单

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拿一条直线

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把中间三分之一替换成一个等边三角形的两条边

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然后对每一段新产生的直线重复同样的操作

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一遍又一遍

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无限重复下去

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最后你得到一片完美的雪花

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这片雪花有一个令人不安的性质

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它围起来的面积是有限的

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但它的边界长度是无限的

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一个装得下的形状

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却有一条你永远走不完的边界

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经典几何学完全没有办法处理这种东西

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六十多年后

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曼德勃罗把科赫雪花和一大批类似的 怪物曲线

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统一到了一个框架里

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一九六七年

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他在 科学 杂志上发表了一篇著名论文

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标题叫

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英国的海岸线有多长

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答案是

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取决于你用多长的尺子去量

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尺子越短

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海岸线越长

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因为每缩小一个尺度

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都会出现新的弯折和凹凸

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你永远量不完

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因为细节在数学上是无限的

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这件事把经典测量学的地基给炸了

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我们一直以为自然界的边界是光滑的曲线

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结果它是粗糙的

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而且那种粗糙在每一个尺度上都长得一样

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海岸线的弯折里藏着更小的弯折

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更小的弯折里还有更更小的弯折

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放大一万倍看

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和放大一倍看

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统计规律完全相同

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曼德勃罗给这种结构起了个名字

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叫分形

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分形的核心特征只有一个词

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自相似

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也就是说

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局部和整体拥有相同的结构

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他发现分形不是数学家的幻想

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而是大自然的默认设计

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而且不只是一种设计

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是最高效的设计

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大自然在用分形解决一个工程问题

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怎么在有限的空间里塞进无限的表面积

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你的肺做了一件非常精妙的事情

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它把支气管像树杈一样分叉了大约二十三级

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每一级的分叉比例几乎一样

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这套分形结构把你胸腔里拳头大小的空间

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展开成了将近一百平方米的气体交换面积

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你的血管系统也是同样的逻辑

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从主动脉到毛细血管

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分形结构让不到一升排量的心脏

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把血液精确送到全身几十万亿个细胞

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一棵树的主干分出大枝

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大枝分出小枝

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小枝分出细枝

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折下一根小枝

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它的分叉形状就像一棵缩小版的整棵树

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河流的水系

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从大江的干流到最细的溪涧

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航拍照片和卫星全景图

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分叉模式几乎一模一样

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闪电的放电路径和你大脑皮层的神经网络分布

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竟然服从同一种分形模式

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甚至你的心跳都是分形的

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健康的心率不是像钟表一样均匀跳动

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而是在每一个时间尺度上都有微小的不规则波动

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心脏病学家发现

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心跳越接近完美均匀

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反而越危险

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因为那意味着心脏失去了跨尺度的自适应能力

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分形

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在某种意义上

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就是生命力本身的数学签名

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这不是巧合

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自然界在用同一套递归算法

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从最小的尺度一直搭建到最大的尺度

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这叫尺度不变性

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不管你把镜头推远还是推近

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看到的模式是同一种

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现在我们回头看法藏说的那句话

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一粒微尘包含整个法界

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如果用分形数学来翻译

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它的意思是

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整体的结构信息

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编码在每一个局部的递归规则里

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你不需要看到整片森林

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才能理解森林的逻辑

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一片叶子的叶脉分形

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已经告诉你了整棵树的生长算法

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一棵树的分叉结构

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和整片雨林的河流水系

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在数学上是同构的

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佛学用四个字说完了这件事

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一即一切

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但分形给了一个更精确的表述

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不是说一等于一切

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而是一和一切共享同一条生成规则

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曼德勃罗集是这件事最极端的证明

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那个边界上最小的芽胞放大之后

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和整个集合的形状完全一致

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不是近似

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不是相似

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是精确一致

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因为它们是同一条公式

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z等于z的平方加c

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在不同初始条件下的输出

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法界缘起要表达的也是这个意思

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华严宗有个核心概念叫事事无碍

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它不是在说万物之间有某种神秘的感应或联系

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而是在说

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万物之所以看起来不同

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只是因为你观察的尺度不同

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换一个尺度看

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同一套法界的结构

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原封不动地出现在每一个现象里

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一朵花和一个星系

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运行的是同一套缘起

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这和因陀罗网有什么区别

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因陀罗网的核心是全息原理

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信息编码在边界上

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局部可以重建全体的信息

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这是一种投影关系

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法界缘起的核心是分形

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局部本身就在运行和全体相同的规则

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这不是投影

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是生成

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打个比方

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一个是说

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你的影子里包含着你的全部信息

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另一个是说

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你身上的每一个细胞里

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都在跑着和你整个身体完全相同的那套基因代码

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全息是信息的等价

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分形是规则的等价

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因陀罗网让你知道

231
00:09:09,383 --> 00:09:12,082
你和宇宙共享同一份数据

232
00:09:12,342 --> 00:09:14,322
法界缘起让你知道

233
00:09:14,342 --> 00:09:17,442
你和宇宙共享同一套源代码

234
00:09:17,849 --> 00:09:20,189
你可以在自己身上验证这件事

235
00:09:20,449 --> 00:09:22,629
下次你看到一棵大树的剪影

236
00:09:22,650 --> 00:09:24,968
你去折下它最小的一根树枝

237
00:09:24,969 --> 00:09:26,189
对着光看看

238
00:09:26,530 --> 00:09:30,149
那根树枝的分叉结构和整棵树几乎一模一样

239
00:09:30,490 --> 00:09:34,068
这就是一即一切最直观的物理证据

240
00:09:34,329 --> 00:09:35,049
不是比喻

241
00:09:35,290 --> 00:09:36,309
不是诗意

242
00:09:36,449 --> 00:09:38,988
是数学结构上的精确对应

243
00:09:39,838 --> 00:09:43,157
曼德勃罗在一九八二年出版了他最重要的著作

244
00:09:43,157 --> 00:09:44,796
叫 大自然的分形几何

245
00:09:45,358 --> 00:09:46,997
他在开篇写了一句话

246
00:09:46,998 --> 00:09:48,816
后来被引用了无数次

247
00:09:49,078 --> 00:09:49,436
他说

248
00:09:49,838 --> 00:09:50,836
云不是球体

249
00:09:51,198 --> 00:09:52,416
山不是锥体

250
00:09:52,478 --> 00:09:54,017
海岸线不是圆弧

251
00:09:54,078 --> 00:09:55,517
树皮不是光滑的

252
00:09:55,797 --> 00:09:57,456
闪电也不走直线

253
00:09:57,838 --> 00:10:00,816
整个欧几里得几何花了两千三百年

254
00:10:00,917 --> 00:10:03,456
描述的全是人类自己造出来的东西

255
00:10:03,758 --> 00:10:06,936
直线 圆 正方形 立方体

256
00:10:07,157 --> 00:10:10,017
这些形状在自然界几乎不存在

257
00:10:10,358 --> 00:10:11,196
真实的自然

258
00:10:11,478 --> 00:10:13,216
从来不用直线思维

259
00:10:13,478 --> 00:10:14,737
它用的是递归

260
00:10:15,078 --> 00:10:16,997
一条规则自己运行自己

261
00:10:17,478 --> 00:10:18,737
一层套一层

262
00:10:18,758 --> 00:10:20,816
直到生成出整个世界

263
00:10:21,078 --> 00:10:22,117
有意思的是

264
00:10:22,117 --> 00:10:24,796
法藏在一千三百年前没有电脑

265
00:10:24,798 --> 00:10:26,057
没有分形数学

266
00:10:26,357 --> 00:10:28,817
但他用了一个极其朴素的物理装置

267
00:10:29,038 --> 00:10:33,376
他在房间的四面墙壁 四个角落 天花板和地板上

268
00:10:33,397 --> 00:10:34,977
全部安装了镜子

269
00:10:35,078 --> 00:10:37,536
中间放一尊佛像和一支火炬

270
00:10:37,678 --> 00:10:40,416
让武则天亲眼看到了无限嵌套

271
00:10:40,678 --> 00:10:43,437
每一面镜子映射所有其他镜子

272
00:10:43,438 --> 00:10:44,536
镜中有镜

273
00:10:44,558 --> 00:10:45,697
像中有像

274
00:10:45,718 --> 00:10:46,557
层层递归

275
00:10:46,798 --> 00:10:47,896
无穷无尽

276
00:10:48,117 --> 00:10:52,457
这就是他能找到的 最接近分形递归的可视化工具

277
00:10:52,888 --> 00:10:55,667
今天你打开任何一个分形渲染软件

278
00:10:55,729 --> 00:10:57,807
输入那条五个符号的公式

279
00:10:57,809 --> 00:10:59,108
然后不断放大

280
00:10:59,128 --> 00:11:02,208
你看到的东西和法藏描述的法界缘起

281
00:11:02,209 --> 00:11:04,068
在结构上是一致的

282
00:11:04,248 --> 00:11:05,128
无限的细节

283
00:11:05,408 --> 00:11:06,767
永远的自相似

284
00:11:06,768 --> 00:11:08,147
局部即整体

285
00:11:08,662 --> 00:11:09,461
这意味着什么

286
00:11:09,902 --> 00:11:12,841
意味着法藏一千三百年前提出的那个

287
00:11:12,902 --> 00:11:15,561
一微尘中现大千世界的说法

288
00:11:15,621 --> 00:11:18,441
在数学上不再是一个疯狂的宣言

289
00:11:18,981 --> 00:11:22,081
它是一个关于递归结构的精确描述

290
00:11:22,662 --> 00:11:23,020
当然

291
00:11:23,422 --> 00:11:24,841
我需要明确地说

292
00:11:24,861 --> 00:11:26,221
这是结构上的同构

293
00:11:26,501 --> 00:11:27,760
不是画等号

294
00:11:28,461 --> 00:11:30,640
分形几何是数学工具

295
00:11:30,662 --> 00:11:33,600
有精确的定义和严格的证明体系

296
00:11:33,942 --> 00:11:37,081
法界缘起是哲学和修行的框架

297
00:11:37,262 --> 00:11:39,921
指向的是对存在本身的洞见

298
00:11:40,381 --> 00:11:43,801
两者的适用范围和证据标准完全不同

299
00:11:44,662 --> 00:11:49,760
但正是这种在完全不同的认知路径上得出的结构趋同

300
00:11:50,022 --> 00:11:53,201
才让人觉得某个东西值得认真对待

301
00:11:53,501 --> 00:11:57,081
一千三百年前有人通过内观得出的洞见

302
00:11:57,142 --> 00:12:01,041
和一九八零年一台电脑屏幕上涌现出来的图形

303
00:12:01,142 --> 00:12:02,380
指向了同一个结论

304
00:12:02,822 --> 00:12:05,301
这个宇宙不是由东西构成的

305
00:12:05,302 --> 00:12:06,941
是由规则构成的

306
00:12:06,942 --> 00:12:08,240
而那条规则

307
00:12:08,302 --> 00:12:11,240
简单到只需要五个符号就能写完

308
00:12:12,158 --> 00:12:14,178
曼德勃罗二零一零年去世

309
00:12:14,198 --> 00:12:15,658
享年八十五岁

310
00:12:15,798 --> 00:12:18,197
他生前接受采访时说过一句话

311
00:12:18,198 --> 00:12:19,378
我印象很深

312
00:12:19,567 --> 00:12:20,368
他说

313
00:12:20,399 --> 00:12:23,097
我一辈子做的事情就是在告诉人们

314
00:12:23,158 --> 00:12:24,817
粗糙不是缺陷

315
00:12:24,958 --> 00:12:27,538
粗糙才是自然界真正的语法

316
00:12:27,678 --> 00:12:29,237
光滑是人类的简化

317
00:12:29,639 --> 00:12:31,258
粗糙才是真相

318
00:12:31,565 --> 00:12:34,625
这和法界缘起有一种奇妙的呼应

319
00:12:35,126 --> 00:12:37,305
佛学的核心洞见之一就是

320
00:12:37,445 --> 00:12:39,944
我们的分别心在把世界简化

321
00:12:40,366 --> 00:12:41,785
我们画出边界

322
00:12:41,925 --> 00:12:45,104
分出你我 内外 大小

323
00:12:45,406 --> 00:12:46,905
但法界缘起说

324
00:12:46,925 --> 00:12:49,064
你画的那些边界不存在

325
00:12:49,406 --> 00:12:51,305
每一层都是全体

326
00:12:51,445 --> 00:12:53,944
每一个尺度都是完整的法界

327
00:12:54,205 --> 00:12:57,425
你不需要走到宇宙的尽头才能看懂宇宙

328
00:12:57,606 --> 00:12:58,885
你此刻站的地方

329
00:12:59,205 --> 00:13:00,824
就已经是全部了

330
00:13:01,657 --> 00:13:05,276
如果宇宙的复杂性真的来自于规则的自我递归

331
00:13:05,417 --> 00:13:07,036
那你是谁这个问题

332
00:13:07,298 --> 00:13:08,757
可能需要重新问

333
00:13:08,978 --> 00:13:10,997
你不是宇宙里的一个零件

334
00:13:11,137 --> 00:13:14,516
你是宇宙在你这个尺度上的完整运行

335
00:13:14,858 --> 00:13:16,717
那条生成星系的规则

336
00:13:16,818 --> 00:13:19,117
此刻也在生成你的念头

337
00:13:19,537 --> 00:13:20,396
想想看

338
00:13:20,458 --> 00:13:23,837
你做的每一个决定都在塑造你的下一个决定

339
00:13:23,938 --> 00:13:26,737
你的每一次情绪反应都在调整

340
00:13:26,737 --> 00:13:29,036
你下一次情绪反应的阈值

341
00:13:29,458 --> 00:13:31,676
你的心理模式是递归的

342
00:13:31,698 --> 00:13:33,997
你的习惯是自相似的

343
00:13:34,098 --> 00:13:36,697
你十年前在关系中犯的错误

344
00:13:36,698 --> 00:13:38,237
和今天犯的那个

345
00:13:38,257 --> 00:13:39,016
放大了看

346
00:13:39,377 --> 00:13:41,197
结构惊人地一致

347
00:13:42,017 --> 00:13:43,757
佛学管这个叫轮回

348
00:13:43,818 --> 00:13:45,757
管这个叫种子熏习

349
00:13:46,098 --> 00:13:47,617
分形数学管这个叫

350
00:13:47,938 --> 00:13:48,636
迭代

351
00:13:49,017 --> 00:13:51,636
同一条公式在不同时间点的输出

352
00:13:52,017 --> 00:13:52,997
而修行

353
00:13:53,058 --> 00:13:55,717
也许就是改写那条公式里的某个参数

354
00:13:55,978 --> 00:13:57,357
哪怕只改了一点点

355
00:13:57,657 --> 00:13:59,556
经过无数次递归之后

356
00:13:59,617 --> 00:14:01,636
生成的图形会完全不同

357
00:14:02,445 --> 00:14:03,004
你怎么看

358
00:14:03,524 --> 00:14:06,403
你觉得自然界的自相似是巧合

359
00:14:06,404 --> 00:14:08,683
还是某种更深的结构在说话

360
00:14:09,284 --> 00:14:10,004
我是王利杰

361
00:14:10,044 --> 00:14:11,091
我们下期见