Prior to the invention of glass, bronze and tin alloy were commonly used as materials for casting mirrors[1]. Due to its unique physical properties, mirrors are closely intertwined with Chinese traditional culture and customs. They play an indispensable role in various aspects of life, such as birth, marriage, funeral rites, medicine, divination, and adornment etc[2].

In traditional rituals, mirrors are often regarded as sacred objects or ceremonial tools to reflect the image of deities[3]. They serve as a medium of communication between humans and the divine during ceremonies. People believe that mirrors can connect the human and divine worlds, allowing glimpses of the future or receiving revelations through their reflection.

From ancient times to the Qing Dynasty, bronze mirrors were essential items in people's lives. The casting of bronze mirrors not only required a smooth surface on the front but also featured various meaningful patterns and even accompanied by poetry on the back for aesthetic purposes. This tradition continued for over two thousand years, resulting in a magnificent collection of ancient Chinese mirrors[4].

[1] National Museum of History Editorial Committee. Collection of Ancient Bronze Mirrors at the National Museum of History (Taiwan, National Museum of History, 1996).

[2] For example, mirrors are used in wedding ceremonies, symbolizing the pursuit of happiness by humans. In funeral rituals, mirrors are used to express humanity's hope for a future life.

[3] In traditional religious ceremonies, mirrors are often placed on altars or shrines to receive the spiritual energy or presence of deities. Worshipers can gaze into the mirror's reflection during worship to establish a connection with the divine. Sometimes, mirrors are also used to ward off evil or dispel negative energies, ensuring the purity of the ritual space.

[4] Shi, Cuifeng. "The Evolution of Chinese Bronze Mirrors." Chinese Art History, National Taiwan Museum of Fine Arts, Taipei (1992), Issue: 26.

The Chinese magic mirror is a special mirror made of bronze in ancient China, representing an astonishing traditional technology of human civilization. Under normal conditions, it appears no different from an ordinary mirror. However, when the mirror surface is illuminated by a strong and concentrated light source, it projects hidden messages, including text or patterns, that are embedded within it. As the magic mirror always reveals the texts and decorations on its backside, creating a phenomenon of light passing through the back of the mirror to the mirror surface, it is called a "translucent mirror." In reality, it is an optical effect.

Based on the existing archaeological findings of magic mirrors, this technology can be traced back to the Western Han Dynasty [5] (202 BC - 195 BC). The earliest written record of this technique can be found in the ancient text " Records of Ancient Mirrors [6]" during the Sui and Tang Dynasties (618-907 AD). Subsequently, magic mirrors appeared during the Western Han, Tang, and Song Dynasties, but the manufacturing methods were not passed down. It was not until the Northern Song Dynasty that the scientist Shen Kuo (1031-1095 AD) provided a detailed description of the physical properties of the magic mirror and his speculation on its casting principles in his work " Dream Pool Essays[7]."

During the Edo period (1603-1867), the Japanese shogunate implemented a policy of national isolation and issued a 'ban on Christianity’, which deemed Christians as practitioners of a heretical religion. As a result, cruel religious persecution took place. In response, Christians utilized the Chinese magic mirror, a translucent mirror, to conceal religious icons and conduct secret worship. Additionally, the magic mirror became a means for believers to recognize each other and evade religious and political persecution.

[5] The earliest existing magic mirror was unearthed in 1982 from an ancient Han Dynasty tomb in Jiutoushan, Liuzhou, China, and is currently stored in the Shanghai Museum.

[6]   The "Records of Ancient Mirrors" written by Wang Du during the Sui and Tang Dynasties is considered to be the beginning of Tang Dynasty novels and holds an important position in Chinese history. It mentions: "Someone once gave him a bronze mirror, and when the sunlight was parallel to the mirror's surface, the patterns on the back of the mirror would be fully displayed in the projection. When the Sui and Tang Dynasties fell, the mirrors also disappeared." Magic mirrors were regarded as legends until the excavation of Western Han artifacts confirmed their existence.

[7] "Dream Pool Essays" is an important document in the history of Chinese science and technology. It covers various advanced achievements in the fields of astronomy, mathematics, physics, geography, medicine, and music, making it a significant work and valuable resource for the study of Chinese scientific and technological history.

During the Tang Dynasty, the great Daoist master Sima Chengzhen(司馬承禎)condensed the Daoist concepts of the universe and all living things into a bronze mirror, which he presented to Emperor Xuanzong(唐玄宗) as a gift on the Qianqiu Festival(千秋節). This mirror served as the prototype for the Eight Trigrams Mirror.[8] Daoism believes that bronze mirrors have the ability to "reflect the prototypes of spirits and demons," making them an important Daoist instrument.

The mirror is circular on the outside and square on the inside, symbolizing the roundness of heaven and the squareness of earth. The mountain-patterned knob, combined with the surrounding mountain patterns, represents the Five Sacred Mountains(五嶽)[9]. Inscribed between the Five Sacred Mountains are sixteen seal-script characters that, when read in sequence, do not form a coherent meaning. Instead, they should be read in a skip pattern[10]. This signifies that this Daoist mirror encompasses the universe and all its beings.

The decorative wave patterns on the outer rim imitate the Four Great Rivers (四瀆)[11]. Within the square-shaped Eight Trigrams, the landscapes of the four directions and the deities of heaven and earth are placed accordingly. The Eight Trigrams are arranged in the sequence of the Eight Trigrams of King Wen (文王八卦), with Qian in the northwest, Kun in the southwest, Li in the south, Kan in the north, Zhen and Dui divided between northeast and southeast. The sun, moon, and stars are depicted outside the Eight Trigrams[12].

This bronze mirror encapsulates the profound Daoist philosophy and the ancient Chinese people's understanding of the interplay between astronomy, geography, and humanities in the world.

[8] Along with presenting the bronze mirror to Emperor Xuanzong, Sima Chengzhen also offered a document titled "Shangqing Hanxiang Jianjian Tu(清寒香鑑鑑圖)" (Diagram of the Supreme Clarity Containing the Sword and Mirror). The mirror gained Emperor Xuanzong's favor and he personally inscribed a poem titled "Reply to Sima Chengzhen on the Sword and Mirror(劍鏡答司馬承禎)."

[9] The Five Sacred Mountains in Daoism, consist of Mount Tai(泰山) in the east, Mount Hua(華山) in the west, Mount Heng(衡山) in the south, Mount Heng(恆山) in the north, and Mount Song(嵩山) in the center. These mountains symbolize the deities of the four directions since the Han Dynasty and are regarded as both the dwelling places of immortals and the sacred sites for cultivating Daoism.

[10] "The universe contains symbols, the sun and moon are steadfast, it records the rules of all things, and enlightens the myriad spirits."

[11] The Four Rivers refer to the Yangtze River(長江), Yellow River(黃河), Huai River(淮河), and Ji River(濟水).

[12] Jiangsu, Yin Zhao, Miao Biyu, 'A Study of Tang Dynasty Sun, Moon, Stars, Five Sacred Mountains, Zodiac, and Eight Trigrams Mirror,' Collectibles Express (2016)

The principle behind the 'Chinese Magic Mirror' involves the optical effect caused by slight variations in curvature on its surface, resulting in the refraction of hidden patterns.

When a ball is thrown onto the ground, it rebounds. Similarly, light exhibits a similar phenomenon during its propagation. The glass of a building reflects light intensely, whereas the rough exterior walls do not. This is due to light being able to concentrate and reflect in a specific direction on a smooth surface, whereas on a rough surface, light scatters in various directions. The dispersed light prevents the exterior walls from reflecting light in a unified direction. The 'Chinese Magic Mirror' operates on this principle to project light with differing degrees of brightness and darkness.

It is widely known that bronze does not possess the physical characteristic of transparency. Thus, when the Chinese magic mirror captured Western attention in 1832, the academic community has been studying the secret to making bronze mirrors translucent[13]. However, it wasn't until a hundred years later that the British scientist William Bragg unraveled the optical principles behind it[14].

[13] This is also the origin of the Western term 'magic mirror' used to describe the ancient Chinese 'translucent mirror’.

[14] The UNESCO Courier, ‘Magic Mirrors’, The Chinese scientific genius, ISSN 0041-5278 (1998), P. 16–17.

According to Joseph Needham, a British biochemist, historian of science, and sinologist, meticulous optical experiments have demonstrated that the "magic" mirrors reproduce the patterns on their backs as a result of minute variations in curvature. The regions with greater thickness exhibit a slightly flatter mirror surface, and in certain instances, they are even concave[15].

The manufacturing process involved casting the basic mirror shape, including the design on the back, in a flat form. Subsequently, the surface was made convex through meticulous scraping and scratching. It was then polished to achieve a shiny appearance. These procedures created stresses that caused the thinner parts of the surface to bulge outward and become more convex than the thicker sections. Finally, a mercury amalgam was applied to the surface, inducing further stresses and selective buckling.

As a result, the imperfections on the mirror surface corresponded to the patterns on the back, although they were too minuscule to be visible to the naked eye. However, when the mirror reflected bright sunlight onto a wall, the entire image became magnified, reproducing the patterns as if they were passing through the solid bronze via beams of light. Sir William Bragg, upon discovering this in 1932, remarked, "Only the magnifying effect of reflection makes them plain." Needham aptly refers to this as "the first step on the road to knowledge about the minute structure of metal surfaces."

[15] The UNESCO Courier, ‘Magic Mirrors’, The Chinese scientific genius, ISSN 0041-5278 (1998), P. 16–17.

In technical terms, the phenomenon observed in the image of the magic mirror can be described as a pre-focal phenomenon. The brightness of the image is determined by the curvature of the surfaces at the top and bottom of the step. A tighter curve results in more focused light, creating a brighter spot, while a looser curve produces a darker spot. The rate at which the curve's gradient changes can be understood as how tightly the line is curved. Mathematically, the rate of change of gradient corresponds to the second derivative of the curve. Plotting this second derivative yields a pattern similar to what we observe in the image reflected by the magic mirror, where bright areas are adjacent to dark areas.

It is worth noting that the manufacturing process of these magic mirrors is closely guarded by only a few manufacturers, who keep their techniques confidential.

The Friedmann equation is a set of equations that describe the model of an expanding universe in a homogeneous and isotropic framework of general relativity. It was first proposed by Alexander Friedmann (1888-1925) in 1922. These equations are obtained by applying Einstein's field equations to the stress-energy tensor of a fluid with a given mass density ρ and pressure p, under the Friedmann–Lemaître–Robertson–Walker metric. The equations for negative spatial curvature were derived by Friedmann in 1924.

The Friedmann equations depend on their constituent elements and provide an explanation for the evolving expansion rate of the universe. The initial Friedmann equation elucidates the temporal variations in the expansion rate of the universe, contingent upon its contents. To understand the origins and future path of the universe, it is sufficient to measure the current rate of expansion and the composition of its contents. Utilizing this equation enables making well-informed forecasts regarding its subsequent development.

Parameter k in the Friedmann equation

The Friedmann equation includes a parameter k that is associated with spatial curvature. This parameter k is used to describe the geometric shape and curvature properties of the universe.

In the Friedmann equation, k can take three possible values: positive (k=1), negative (k=-1), and zero (k=0).

These values correspond to three possible geometric shapes of the universe: a closed universe (positive curvature, spherical), an open universe (negative curvature, hyperbolic), and a flat universe (zero curvature, planar).

Thus, the definition and values of k in the Friedmann equation are significant in describing the geometric structure and expansion behavior of the universe, providing important insights into the evolution and future destiny of the universe.

Nearly one hundred years following the publication of his most renowned work, Friedmann's equations have been expanded to encompass a Universe characterized by an inflationary origin, dark matter, neutrinos, and dark energy. Remarkably, these equations remain entirely applicable, necessitating no additional amendments or adjustments to accommodate these remarkable advancements. They establish a connection between the existing matter and energy within the Universe and its present, past, and future expansion rates, thereby providing insights into the fate and historical trajectory of the Universe based on contemporary measurements.