International Journal on Architectural Science, Volume 9, Number 1, p.1-17, 2012

DISCOVERY OF A GEO-GEOMETRIC WONDER OF THE WORLD : THE PEAK-HARBOUR LAYOUT OF HKSAR WITH REDEFINING THE WESTERN BOUNDARY OF OF HKSAR

K.P. Cheung Department of Architecture, The University of , Hong Kong, China

(Received 21 January 2013; Accepted 5 March 2013)

ABSTRACT

Hong Kong Special Administrative Region of China, HKSAR, consists of HK Island (represented by ), and , (i.e. N.T., represented by ), (represented by , or Phoenix Peak - translation of the Chinese name), Victoria Harbour (bounded by Pass to the East, and The West Boundary joining Tsing Yi Island, HK Island, and Ting Kau of N.T.) and some other islands and sea regions.

Four straight lines joining the highest natural points of Victoria Peak, Lion Rock, Lautau Peak [i.e. Phoenix Peak], and one point at the Existing Eastern Boundary Line and another point at the Proposed Western Boundary Line of Victoria Harbour, will form a world-iconic Six-Point Four-Line Geometric figure. The sixth point is the intersecting point of two of the four geometric lines, and this sixth point coincides with the central position of the existing International Commerce Centre at West Kowloon, which is the tallest building in HKSAR. Applying Menelaus' Theorem directly, an established Euclidian Plane geometric theorem, will result in obtaining “1” on multiplication of ratios of the map distances between the Peaks, the two points at East and West boundaries of Victoria Harbour, and International Commerce Centre at West Kowloon, provided the existing western boundary is shifted from its existing location at central-south of Tsing Island to the South- western end of Tsing Yi Island, joining to Green Island, as proposed in this paper, resulting in a slight expansion of Victoria Harbour.

Using all the same six points as above and adding two more lines which intersect, a Seven-Point Six-line geometric figure is formed. Applying Ceva’s Theorem to this geometric figure, which is also an established Euclidian Plane geometric theorem, will also result in obtaining “1” on multiplication of ratios of the map distances.

Thus the two theorems will produce Two “1”s, hence Double-One, world legacy for HKSAR in physical geo- geometric sense.

This “Double-One” discovery will be accomplished, if the Western Boundary of Victoria Harbour is shifted to touch onto the South-Western end of Tsing Yi Island, by expanding the coverage of Victoria Harbour westward, and hence protecting a larger harbour.

While mountains and peaks exist in nature without human intervention to define their height, the harbour boundary lines are artificially determined. Attaining world-iconic Geo-Geometric records, and simultaneously protecting a slightly expanded harbour, contributing to world mathematical education and geographical studies, and boosting local and international tourism on actual site visits to HKSAR or via internet, are overriding reasons for Redefining the Western Boundary of Victoria Harbour.

1. INTRODUCTION world famous Victoria Harbour (bounded by Lei Yue Mun Pass at East [21, Fig. 5a, Fig. 5b] and Hong Kong Special Administrative Region of The West Boundary joining Tsing Yi Island, HK China, HKSAR [1] consists of HK Island Island, and Ting Kau of N.T.) [Fig. 6a, Fig. 6b], (represented by Victoria Peak of 552 m high [2, and some other islands and sea regions. These Fig. 1, Fig. 2]), Kowloon and New Territories (i.e. islands and sea regions constitute the entire N.T., represented by Lion Rock of 495 m high [3, HKSAR [1,6, Fig. 7]. Fig. 3a, Fig. 3b ]), Lantau Island (represented by Lantau Peak, i.e., Phoenix Peak of 934 m high, [4, The top parts of Victoria Peak, Lion Rock, and Fig. 4] the translation of the Chinese name), the Lantau Peak [i.e. Phoenix Peak] are physically

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defined in nature by its existence above ground, changed. The Western harbour boundary of and are not supposed to be changed by humans. Victoria Harbour currently formed by the line connecting the West Part of Green Island and the Belonging to the same “Kowloon Granite” group as South-Central part of Tsing Island can be shifted Lion Rock [5,7], Lei Yue Mun Pass [21], inheriting for overriding good reasons, as discussed in Section a unique geographical Pass Form guarding the 5 [8, Fig. 6, Fig. 8, Fig. 10, Fig. 11]. Eastern Boundary of Victoria Harbour, shall not be

Fig. 1: Victoria Peak [552 m high] in the background, , and 2 International Finance Centre, 412 m tall, in the foreground. Photo of 2 August 2008, Courtesy of Dr. Sam C M Hui, Department of Mechanical Engineering, The

Fig. 2: The top of Victoria Peak, 552 m high, with the radio stations, the most well-known peak on Hong Kong Island [2]. Photo courtesy of Wikipedia.org

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Fig. 3a: Lion Rock, 495 m high, guarding and New Territories [3]. Photo courtesy of Wikipedia.org

Fig. 3b: Lion Rock, 495 m high, guarding Kowloon Peninsula on the left and New Territories on the right [3], photo taken from the East Side of Lion Rock. Photo Courtesy of Agriculture, Fisheries and Conservation Department, HKSAR

Fig. 4: Lantau Peak, i.e., Phoenix Peak, 934 m high [4], the most well-known peak on Lantau Island, the traditional favourite place for watching sunrise. Photo Courtesy of Hong Kong Tourist Association, HKSAR

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Fig. 5a: Lei Yue Mun, taken from the shore of Sam Ka Tsuen, Lei Yue Mun, viewing towards East [21]. Photo courtesy of Wikipedia.org

Fig. 5b: Lei Yue Mun, taken from of Hong Kong Island, viewing towards North [21]. Photo courtesy of Wikipedia.org

Fig. 6a: The existing coverage and boundaries of Victoria Harbour, HKSAR, governed by Protection of the Harbour Ordinance [8]. East Harbour Boundary is at Lei Yue Mun Pass, and West Boundary is connected to Tsing Yi Island. Victoria Peak is on the Left side of Hong Kong Island. Photo Courtesy of Harbour-front Enhancement Committee, HKSAR

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Fig. 6b: Victoria Harbour of HKSAR: International Commerce Centre, ICC, 490 m tall in West Kowloon in the background at the Left of photo, 2 International Finance Centre, 412 m tall on Hong Kong Island, at Right of Photo, with Victoria Harbour at the middle, and Lei Yue Mun Pass, the Eastern Boundary of Harbour is hidden around Top Right of Photo. Photo courtesy: Hong Kong Tourist Association

Fig. 7: Map of Hong Kong Special Administrative Region. Photo courtesy of Lands Department, HKSAR [6]

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2. THE WORLD-ICONIC FOUR-LINE 4. A line joining “Point EBVH” at Lei Yue Mun GEOMETRIC PEAK-HARBOUR Pass to ICC [490 m tall, “Point ICC”, West LAYOUT OF HKSAR Kowloon] will be extended to meet “Line PP- LR” at “point WBVH” lying on the The steps for drawing the Four-Line Geometric PROPOSED NEW Western Boundary of Peak-Harbour Layout of HKSAR are as follows: Victoria Harbour. “Line EBVH-ICC-WBVH” [Fig. 8, Fig. 9, Fig. 10] will be formed. “Line PP-LR” of Step 2 will become “Line PP-WBVH-LR”. 1. A line joining the top of Victoria Peak [552 m high, “Point VP”] to Lion Rock [495 m high, 5. IT IS PROPOSED that the NEW Western “Point LR”] will pass through the centre of Boundary of Victoria Harbour to be shifted ICC [490 m tall, West Kowloon, “Point ICC”, to the West, on a line joining the existing Fig. 7, Fig. 9], forming “Line VP-ICC-LR”. point at Green Island of Victoria Harbour Boundary and the New “Point WBVH” 2. A line joining the top of Lion Rock [495 m proposed in Step 4, extended to touch onto the high, “Point LR”] to Lantau Peak [Phoenix western tip of Tsing Yi Island. Peak in Chinese, 934 m high, “Point PP”] will form a line called “Line LR-PP” passing Other boundary lines between Green Island and through the Western Boundary of Victoria HK Island, between Tsing Yi Island and Ting Kau, Harbour near Tsing Yi Island, at a point called N.T. will remain UNCHANGED. “Point WBVH” [Fig. 10]. [Note: This line will be called “Line LR-WBVH-PP” after Drawing these 4 lines over HKSAR linking the performing Step 4 below.] prime three peaks and Victoria Harbour and using the proposed new Western Boundary of Victoria 3. A line joining the top of Lantau Peak Harbour will form a world-iconic and unique Four- [Phoenix Peak, 934 m high, “Point PP”] to line Geometric Figure, hinging onto International Victoria Peak [552 m high, “Point VP”] will Commerce Centre at West Kowloon, 490 m tall, be extended to meet the Eastern Boundary of The Silver Dragon Building [9-12]. Victoria Harbour at Lei Yue Mun Pass at “Point EBVH” [Fig. 8, Fig. 10].

Fig. 8: Hong Kong has a Legend: The Six-Point Four-Line Geo-Geometry of Peak-Harbour Layout HKSAR forms the basis of Discovery of A Geo-Geometric Wonder of the World: The Peak-Harbour Layout of HKSAR with Redefining the Western Boundary of Victoria Harbour of HKSAR [11]. Please see text on the steps of formation of the geometric figure.

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Fig. 9: A line joining the top of Victoria Peak, 552 m high, “Point VP”, to Lion Rock of 495 m high, “Point LR”, will pass through the centre of International Commerce Centre of 490 m tall, West Kowloon, “Point ICC”, the tallest building in HKSAR, forming “Line VP-ICC-LR” [9]. The background drawing is a survey drawing of scale of 1:5000, produced by Lands Department, HKSAR. See Fig. 8, Fig. 10, Fig. 11 on nomenclature of the “Points”.

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3. THE GEO-GEOMETRIC WONDER Applying another famous well established OF THE WORLD: THE DOUBLE– theorem, Ceva's theorem, 施華氏定理 [13,15,16, ONE GEOMETRIC PEAK- Appendix 2, Fig. 14] to this unique 7-Point 6-Line HARBOUR LAYOUT OF HKSAR Peak-Harbour geometry of HKSAR, the second “One” is obtained: [ Fig. 11] Application of Menelaus' Theorem (Note: It should be noted that the labeling of points 萬年樂善氏定理 [Fig. 10, Fig. 13] in Fig. 10 and Fig. 11 is different in order to maintain consistency with Fig. 13 [Appendix 1] Applying the famous well established Menelaus' and Fig. 14 [Appendix 2] respectively.) Theorem 萬年樂善氏定理 [Fig. 10, Fig. 13] to this unique 6-Point 4-Line Peak-Harbour geometry The Ratio of lengths (i.e. distances) = (AF/FB) x of HKSAR, the First “One” is obtained: [Fig. 8, (BD/DC) x (CE/EA) = 1 [Fig. 11, Fig. 14, Fig. 10] Appendix 2]

The Ratio of lengths (i.e. distances) = (AF/FB) x = [length of line PP-VP / length of line VP- (BD/DC) x (CE/EA) = 1 [Fig. 10, Fig. 13, 14, 16, EBVH] x [length of line EBVH – Point Ceva / Appendix 1] length of line LR- Point Ceva] x [length of line LR-WBVH / length of line WBVH-PP] = [length of line PP-EBVH / length of line EBVH- VP] x [length of line VP-ICC / length of line ICC- = [23.3 km / 9.5 km] x [3.9 km / 4.9 km] x [10.2 LR] x [length of line LR-WBVH / length of line km / 19.6 km] WBVH-PP] = 1.016 = 1 [Fig. 11] = [32.8 km / 9.3 km] x [3.4 km / 6.2 km] x [10.2 km / 19.6 km] (The deviation from “1” is due to measurement error when scaling the actual distances from the = 1.007 = 1 [Fig. 10] 1:50000 map and initial round up on numbers.)

(The deviation from “1” is due to measurement This is the Second geometric “One” of world error when scaling the actual distances from the wonder, obtained by applying Ceva’s theorem, 1:50000 map and initial round up on numbers.) named in honour of Giovanni Ceva (AD 1647 – 1734) [15]. This is the First geometric “One” of world wonder, obtained by applying Menelaus' Theorem, named Now HKSASR will score “Double-One” on 6- in honour of Menelaus of Alexandria (c. 70–140 Point 4-Line & 7-Point 6-Line Geo-Geometric CE) [14]. Peak-Harbour Layout of HKSAR, which is world-unique. The first “One” is obtained by Application of Ceva’s Theorem 施華氏定理 applying Menelaus' Theorem on the 4-Line [Fig. 11, Fig. 14] Geometric figure, and the second “One” is obtained by applying Ceva’s Theorem on the 6- Adding two more lines, namely from point EBVH Line geometric figure. This Geo-Geometric to point LR and from point PP through ICC to the Wonder of the World will be fully realized when intersection with line EBVH-LR, to the basic the proposed new western boundary of Victoria unique 6-Point 4-Line geometric Peak-Harbour Harbour is put into legislation and enactment. Layout of HKSAR will produce a unique 7-Point 6-Line geometric Peak-Harbour Layout of the HKSAR.

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Fig. 10: The Geo-Geometric Wonder of the World: Applying the famous well established Menelaus' Theorem 萬年樂善氏定理 [13,14,16, Appendix 1, Fig. 13] to this unique 6-Point 4-Line Peak-Harbour geometry of HKSAR, the first “One” is obtained on multiplication of the ratios [See text, Fig. 8, Fig. 10]. The “Points” are highlighted, and the corresponding points of Fig. 13 [Appendix 1] are put into brackets for easily comparison of Fig. 10 and Fig. 13. The background map is of scale of 1:50000, produced by Lands Department, HKSAR, with subsequent lines drawn and measured on it by using steel rules and ball pens, and physical weaving colour lines attached on the map for photo taking. The two closely drawn NNW-SSE running lines touching South-western end of Tsing Yi Island, passing through Point WBVH (E), define the location of the new proposed Western Boundary of Victoria Harbour, and the final new boundary line is subject to further accurate land-sea survey confirmation.

Note that: - VP is Victoria Peak [552 m high] - PP is Phoenix Peak or Lantau Peak [934 m high] - LR is Lion Rock [495 m high] - EBVH is a Point on Eastern Boundary of Victoria Harbour at Lei Yue Mun Pass - WBVH is a point on the newly proposed Western Boundary of Victoria Harbour near Tsing Yi Island - ICC is International Commerce Centre at West Kowloon, 490 m tall

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Fig. 11: The Geo-Geometric Wonder of the World: Applying the famous well established Ceva’s Theorem 施華氏定理 [13,15,16, Appendix 2, Fig. 14] to this unique 7-Point 6-Line Peak-Harbour geometry of HKSAR, the second “One” is obtained on multiplication of the ratios [See text, Fig. 8, Fig.10, Fig. 11]. The “Points” are highlighted, and the corresponding points of Fig. 14 [Appendix 2] are put into brackets for easily comparison of Fig. 11 and Fig. 14. The background map is of scale of 1:50000, produced by Lands Department, HKSAR, with subsequent lines drawn and measured on it by using steel rules and ball pens, and physical weaving colour lines attached on the map for photo taking. The two closely drawn NNW-SSE running lines touching South-western end of Tsing Yi Island, passing through Point WBVH (E), define the location of the new proposed Western Boundary of Victoria Harbour, and the final new boundary line is subject to further accurate land-sea survey confirmation.

Note that: - VP is Victoria Peak [552 m high] - PP is Phoenix Peak or Lantau Peak [934 m high] - LR is Lion Rock [495 m high] - EBVH is a Point on Eastern Boundary of Victoria Harbour at Lei Yue Mun Pass - WBVH is a point on the newly proposed Western Boundary of Victoria Harbour near Tsing Yi Island - ICC is International Commerce Centre at West Kowloon, 490 m tall - D is the point, called Pint Ceva, drawn on the map according to Ceva’s Theorem [15, Fig. 14, Appendix 2] - Point Ceva exists at sea shore of Bypass, proposed to be zoned into Kai Tak District for setting up Ceva’s statue inside Kai Tak District

4. THE PROBABILITY OF iconic, and unique, a geo-mathematical wonder of EXISTENCE OF WORLD-ICONIC 6- the world, a result of co-existence of the following POINT 4-LINE & 7-POINT 6-LINE features: [Fig. 8, Fig. 10, Fig. 11, Fig. 13, Fig. 14] GEOMETRIC PEAK-HARBOUR LAYOUT OF HKSAR, A GEO- - The three region-representative peaks, namely, Victoria Peak, Lion Rock, Lantau Peak (or GEOMETRIC WONDER OF THE Phoenix Peak), and Victoria Harbour exist in WORLD nature, and they are important to HKSAR people since cultures, daily life and economic The existence of this 6-Point 4-Line & 7-Point 6- developments, political and historical changes, Line Peak-Harbour Layout of HKSAR is world-

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all happen below these peaks and around the joining the Eastern Boundary of Victoria harbour; and Harbour at Lei Yue Mun Pass, which is a natural harbour pass, and the proposed new - Collectively the five points, which are the top West Boundary at a point on which the ENE- point of the three region-representative peaks, WSW running line joining Lion Rock, namely, Victoria Peak, Lion Rock, Lantau Peak [representing Kowloon and N.T.] and Lautau (or Phoenix Peak), and the representative East Peak [i.e. Phoenix Peak, representing Lantau boundary point and West boundary point of Peak and International Airport] Victoria, represent HKSAR; and also passes through. This linking relationship can be considered according to line concept - ICC, the man-made building, situated at the 6th consideration of architecture and city planning point of the 6-Point 4-Line & 7-Point 6-Line as a MEANS of uniting the two peaks, and Peak-Harbour Layout of HKSAR, is also their associated lands and Victoria Harbour in significant as it is currently the tallest building primarily E-W direction, thus UNITING in HKSAR, a commercial building, like a Lantau Island, the water regions outside and leader of buildings, leading HKSAR to inside Victoria Harbour, Kowloon, N.T.; and commercial sustainability and prosperity; Regionally ICC acts as the centre feature for - Based on the above N-S and E-W city the re-development of West Kowloon; and connection concepts, and with the existence of ICC, and the Four-line Geo-geometric - Two straight lines, and one extended line formation, the land and the sea of HKSAR are joining the three peaks, pass upon Victoria considered UNITED AS ONE BODY in all Harbour, among which one line only passes principal directions of N-S-E-W (the western upon the water of Victoria Harbour, and the way of expression) or E-S-W-N (the Chinese other two lines pass upon the sea water of way of expression), according to line concept Victoria Harbour and the sea water outside the consideration of architecture and city planning; Harbour boundaries; and and

- The straight line joining the top two most - In the six points of this basic Six-Point Four- important peaks in HKSAR, namely Victoria line Geo-geometric formation, five of which Peak and Lion Rock passes through the Centre are on prime features of HKSAR existing in of the tallest building in HKSAR, symbolically nature, and the newest one is artificial, being the leader of all buildings and man-made the tallest building in HKSAR, International features in HKSAR, The International Commerce Centre, 490 m tall, completed and Commerce Centre, ICC, located in West occupied before the date of this world-wonder Kowloon; and geo-geometric discovery came to the mind of the author on 17 Jan 2012; and - The reclaimed land in West Kowloon developed over the past 30 years enabled ICC - This geo-geometric wonder covers an area of to be built in its current position [Fig. 9]: a huge extent over land and sea bounded BY a process completely led through by God, or with distance of 33 km [from Phoenix Peak, or some human intention? and Lantau Peak to Lei Yue Mun Pass at Eastern Boundary of Victoria Harbour] TIMES a - ICC, 490 m tall, the tallest building in HKSAR distance of 9 km [from Lion Rock to Lei Yue [Fig. 9], situated on the NNE-SSW running Mun Pass at Eastern Boundary of Victoria line joining two most important peaks in Harbour]; and HKSAR, namely Victoria Peak and Lion Rock, which respectively leads Hong Kong Island, - In the sense of representation of the entire and Kowloon and New Territories, and on its HKSAR by the THREE Prime peaks and the way also goes through the land of Hong Kong Prime Harbour of HKSAR, the reported Geo- Island, the water of Victoria Harbour, the land Geometric wonder of the world embraces the of Kowloon Peninsula, can be considered entirety of HKSAR according to line concept consideration of architecture and city planning as a MEANS of It is guessed that the calculated probability of the uniting the two peaks, and their associated combinations of happening of co-existence of the lands and Victoria Harbour in primarily N-S above features and their inter-relationships will be direction,, thus UNITING HK Island-Victoria extremely small, implying that this discovered Harbour-Kowloon-N.T.; and world-wonder is of extremely high value, making HKSAR not only “Pearl of the Orient”, rather - ICC, the tallest building in HKSAR [Fig. 9], is “Pearl for the World”. However this is the topic of situated along the ESE-WNW running line another paper.

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5. FURTHER EXPLORATION ON ICC, the “Menelaus Point”, can be equipped at the THE DISCOVERY “Sky 100 Floor” viewing floor, 393 m above ground, with telescopes, maps, laser lighting The reported geometric figures [Fig. 10, Fig. 11] devices, and other audio-visual aids to help visitors are associated with a wide range of established to understand the various points, features, and lines theorems and related cases and illustrations [13- of this discovered geo-geometric wonder of the 17], within and beyond plane geometry, which can world. be further considered and applied to this Geo- geometric discovery, for exploring on further In Menelaus Park at West Kowloon [Fig. 12a, findings and meaning of this discovered basic 6- 12b], the statue of Menelaus of Alexandria [14] Point 4-Line & 7-Point 6-Line Peak-Harbour should be erected, facing Alexandria of Egypt, the geometric layout in HKSAR. home town of Menelaus. Likewise in Ceva Park in Kwun Tong, which should be zoned within Kai Subject to further investigation and confirmation, Tak District for excelling this discovery, where this discovery possibly is the largest geographic “Ceva point”, in Chinese, “施華氏點”, is located existence of 6-Point 4-Line & 7-Point 6-Line [Fig. 4], the statue of Ceva of Alexandria should geometric figures, linking points of physical, be erected, facing Milan of Italy, the home town of cultural, and city significance, in which at least Ceva [15,18]. two well established geometric theorems can well be applied and illustrated. This discovery therefore Besides geometric lines, solar movement over is of extremely high value of mathematical seasons and hours will cause International education not only in HKSAR, but worldwide. Commerce Centre, ICC, and Phoenix Peak (i.e. Lantau Peak) to cast solar shadows and brightness This discovery can be called “World Menelaus- contrasting spots onto a large Proposed Mega- Ceva Geo-Geometric Wonder”, covering an area event Plaza which is nearby International of land and sea of about 30 km x 9 km, a Geo- Commerce Centre at West Kowloon, adjacent to Geometric physical wonder of the world, in the the High Speed Rail Station [11,19,20, Fig. 12a]. famous city traditionally named “Pearl of the International Commerce Centre and the Proposed Orient”, now to be also called “Pearl for the Mega-event Plaza will form The Silver Dragon World”, and possibly “Menelaus-Ceva City”, in Sundial [Fig. 12a, Appendix 3], while Phoenix honour of Menelaus and Ceva whose names are Peak [i.e. Lantau Peak] and the Proposed Mega- conferred to the theorems illustrated in HKSAR. event Plaza will form the Golden Phoenix Sundial. These two sundials are world-iconic sundials by Then where is the Pearl? Was it in the location of themselves, by possessing the tallest man-made the Menelaus Point, which is the location of sundial gnomon of ICC, 490 m tall, and the tallest International Commerce Centre, ICC, the tallest natural sundial gnomon of 934 m tall, the Phoenix building in HKSAR, and the only man-made Peak [also known as Lantau Peak], coupling with feature in this basic 6-Point 4-Line Geo-Geometric the Mega-event Plaza, which is common sundial Peak-Harbour Layout. This is an area where surface for both sundial, receiving the solar science meets cultures, histories, political, gnomonic shadows and heavenly solar brightness economic and city developments, an integrated and prosperity. Furthermore since Phoenix Peak is area for further investigation. about 26 km from the Mega-event Plaza, the Golden Phoenix Sundial will be the largest sundial The location of ICC, can be called “Menelaus in the world [Fig. 12a, Fig. 12b]. And the Silver Point”, in Chinese, “萬年樂善氏點” [Fig. 10, Fig. Dragon Sundial will become the world-iconic 11], and perhaps ICC be given another name Menelaus Silver Dragon Sea-Land Sundial “Menelaus Tower”, in honour of Menelaus of 萬年樂善銀龍海陸日晷 [Fig. 9, Fig. 12a, Fig. Alexandria. The park surrounding “Menelaus 12b, Appendix 3]. Point” at West Kowloon, part of which will overlap with West Kowloon Cultural District, shall Other considerations will include, but not limited be developed into “Menelaus Park” [Fig. 12a. to, the processes of shifting the current West 12b], and the city of HKSAR can be considered to boundary of Victoria Habour by adopting the establish sister-city relationship with Alexandria in proposal of this paper, installations of sufficient Egypt, in honour of Menelaus of Alexandria (AD lighthouses for harbour boundary identification and 70–140) [14]. This will be a contribution of for enhancing sea travelling safety; and obtaining architecture, city planning, international accurate surveyed distances of the lines of this friendship, tourism, to mathematics, and vice reported 6-Point 4-Line & 7-Point 6-Line Geo- versa. Geometric Peak-Harbour layout, using land survey methods, and GPS methods.

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Fig. 12a: Menelaus Park 萬年樂善公園: Eastern portion to be developed into The Proposed Mega-event Plaza [enclosed area in Blue dotted lines], containing World-Iconic Dual Sundial Surface, namely The Silver Dragon-Mega Event Plaza Sundial, and The Golden Phoenix-Mega Event Plaza Sundial [9- 12,19,20, Appendix 3]. The area bounded by the Red dotted lines is the West Kowloon Cultural District. The Silver Dragon Sundial will become the world-iconic Menelaus Silver Dragon Sea-Land Sundial 萬年樂善銀龍海陸日晷 [Appendix 3].

Fig. 12b: Menelaus Park: The overall concept -The vicinity of ICC, the “Menelaus Point”, i.e. the 6 Red geometric lines radiating out from the Menelaus Point, should be landscape or roof features to excel this world-wonder straight geometric pattern, to be clearly viewed from satellites and air crafts as an extensive and straight world-iconic pattern, bearing cultural and ecological riches. The Menelaus point and these theorem-excelling features can be collectively into the world-iconic “Menelaus Park” 萬年樂善公園, which should contain the existing West Kowloon Cultural District (WKCD). WKCD should be redesigned to accommodate these geometric features and the relating viewing and celebrating provisions, and the Mega-event Plaza to complementarily and synergistically excel this discovery of Geo- Geometric Wonder of the World: The Peak-Harbour Layout of HKSAR with Redefining the Western Boundary of Victoria Harbour of HKSAR [Fig. 12a, Appendix 3].

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6. CONCLUSION 4. Lantau Peak, HKSAR. http://en.wikipedia.org/wiki/Lantau_Peak Starting with constructing the basic Euclidian 6- 5. Supervolcano discovered in Hong Kong. Friday, Point 4-Line plane geo-geometry and applying August 31, 2012, 5:28. Menelaus’ Theorem to it, and then applying http://beforeitsnews.com/earthquakes/2012/08/supe Ceva’s Theorem to the 7-Point 6-line geo- rvolcano-discovered-in-hong-kong-2444398.html geometry, this paper reports on the discovery of A 6. The map of Hong Kong Special Administration Geo-Geometric Wonder of the World : The Region. Land Department, HKSAR Government. “Double–One” Geo-Geometric Peak-Harbour http://www.landsd.gov.hk/mapping/tc/paper_map/t Layout of HKSAR, world-iconic, and unique, p.htm ; which will be realized, when the proposed new http://www.landsd.gov.hk/mapping/en/lic/image/m western boundary of Victoria Harbour is put into ap/hm100cl_front.jpg legislation and enactment. 7. 港九多處花崗岩形成與古代超級火山有關. Radio Hong Kong News. 2012-08-30 HKT 18:03. This “Double-One” discovery will be Video. accomplished, if the Western Boundary of Victoria http://app1.rthk.org.hk/rthk/news/videonews/index Harbour is shifted to touch onto the South-Western _2010.php?news_id=866203&page_no=3 end of Tsing Yi Island, by expanding the area of 8. Protection of the Harbour Ordinance. Victoria Harbour westward, i.e., protecting a larger http://en.wikipedia.org/wiki/Protection_of_the_Har harbour, which should also be a contribution to bour_Ordinance environmental protection. 9. K.P. Cheung, Applying traditional Chinese core values originated from I Ching (1100 BC) in While mountains and peaks exist in nature without conserving the Government House and human intervention to define their height, the Government Hill, and developing HKSAR into a harbour boundary lines are artificially determined. green metropolitan city of harbour - Parks, Overriding reasons for redefining the Western cultures, sundials, underground shopping city, Boundary of Victoria Harbour include attaining world heritage, mega events. world-iconic Geo-Geometric records, the 君子以「自強不息、厚德載物」保育「禮賓府 generosity of sharing with people worldwide on 政府山」建設「香港 : 綠化海港公園之都、文 this discovery, for fostering international 化日晷世寶之都、地下城商場之都、盛事之 friendship, simultaneously protecting a slightly 都」, pp. 16-17,46,47. expanded harbour, contributing to world http://icee.hku.hk/activities/HKU100exhibition/doc mathematical education and geography studies, and /20110316g-101/KPC.pdf boosting local and international tourism on actual 10. 無限風光在香江 Hong Kong has a legend. site visits to HKSAR or via internet. http://icee.hku.hk/activities/HKU100exhibition/doc /20110316g-201/KPC.pdf In addition, The Silver Dragon Sundial, and the 11. K.P. Cheung. Proposals of future developments for Golden Phoenix Sundial will be world-leading HKSAR, 1, 2, 3. architectural and natural sundials in the world, http://icee.hku.hk/activities/HKU100exhibition/doc adding great attraction to tourists and nurture of /20110316g-1/KPC.pdf international friendship. 12. Official Web site of International Commerce Centre. http://www.shkp-icc.com/website/Main.do These sundials by themselves would be the topic of a separate paper, integrating science and art, to 13. W. Cherowitzo. Lecture on higher geometry, pp. be fruitfully discussed by incorporating this 11-14. http://www-math.ucdenver.edu/~wcherowi/ Discovery of Geo-Geometric Wonder of the courses/m3210/lecture1.pdf World : The Peak-Harbour Layout of HKSAR 14. Menelaus' theorem. with Redefining the Western Boundary of http://en.wikipedia.org/wiki/Menelaus%27_theore Victoria Harbour of HKSAR. m 15. Ceva's theorem. http://en.wikipedia.org/wiki/Ceva's_theorem REFERENCES 16. 射影幾何-完全四邊形的調和性質. http://tw.myblog.yahoo.com/oldblack- 1. Hong Kong. wang/article?mid=5308&prev=5317&next=5285& http://en.wikipedia.org/wiki/Hong_Kong l=f&fid=11 2. Victoria Peak, Hong Kong Island, HKSAR. 17. A.D. King and W.K. Schief, “Tetrahedra, http://en.wikipedia.org/wiki/Victoria_Peak octahedra and cubo-octahedra: integrable geometry 3. Lion Rock, Kowloon, HKSAR. of multi-ratios”, Journal of Physics A: http://en.wikipedia.org/wiki/Lion_Rock Mathematical and General, Vol. 36, No. 3, pp. 785- 802 (2003).

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http://people.bath.ac.uk/masadk/papers/multiratio.p APPENDIX 2: CEVA'S THEOREM df [abstracted from 15] 17. Giovanni Benedetto Ceva. http://www- history.mcs.st-andrews.ac.uk/Biographies Ceva's theorem [15,18], named in honour of /Ceva_Giovanni.html Giovanni Ceva, (1647 –1734) is a theorem about 18. K.P. Cheung. The proposed mega-event plaza, triangles in Euclidean plane geometry. Given a containing world-iconic dual sundial : Silver triangle ABC, let the lines AO, BO and CO be Dragon Sundial - Phoenix Sundial Plaza. drawn from the vertices to a common point O to http://icee.hku.hk/activities/HKU100exhibition/doc meet opposite sides at D, E and F respectively. /20110316d-3/KPC.pdf Then, using signed lengths of segments, 19. K.P. Cheung, 香港：綠化之都、文化世寶之 都、盛事之都-首期核心區項目建議書, 第一版, (AF/FB) x (BD/DC) x (CE/EA) = 1 2012 年5 月22 日; HKSAR: A green metropolitan city of cultures, world heritage, mega events : Phase 1 proposals in the core districts, 1st edition, 22 May 2012, first internet publication: 20 June 2012. http://icee.hku.hk/activities/HKU100exhibition/doc /20110316e-1/KPC.pdf 20. Lei Yue Mun. http://en.wikipedia.org/wiki/Lei_Yue_Mun

Fig. 14: Ceva's theorem - A case illustrated with the three lines concurrent at a point O inside APPENDIX 1: MENELAUS' THEOREM ABC [abstracted from 14]

Menelaus' theorem, [14] named in honour of Menelaus of Alexandria, (AD 70–140) is a theorem about triangles in plane geometry. Given a triangle ABC, and a transversal line that crosses BC, AC and AB at points D, E and F respectively, with D, E, and F distinct from A, B and C, then using signed lengths of segments,

(AF/FB) x (BD/DC) x (CE/EA) = 1

or simply AF x BD x CE = FB x DC x EA

Fig. 13: Menelaus' theorem – A case illustrated with line EDF passes inside triangle ABC

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APPENDIX 3: THE SILVER DRAGON SUNDIAL [abstracted from 9, 10, 19, 20]

The Menelaus Sundial (since ICC stands on Menelaus Point), the Silver Dragon Sundial, the world-iconic Menelaus Silver Dragon Sea-Land Sundial 萬年樂善銀龍海陸日晷

Fig. 15a: Photo taken at 16:08, 28 June, 2012 Fig. 15b: Photo taken at 16:18, 28 June, 2012

Fig. 15c: Photo taken at 16:26, 28 June, 2012 Fig. 15d: Photo taken at 16:35, 28 June, 2012

The solar shadow of ICC touching the Fire Station Building at Canton, which will be relocated

Fig. 15a-d: The Silver Dragon Sundial : Photos of the Solar shadow of International Commerce Centre, designed on a rising Dragon concept, taken at “Floor-Sky 100”, 393 m tall, of ICC, 490 m tall, West Kowloon, on 28 June, 2012, with the solar shadows cast onto the Plaza adjacent to High Speed Rail Station, [Fig. 12] at four time slots, namely 16:08, 16:18, 16:26, 16:35.

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Fig. 16: The Silver Dragon Sundial : Photos of the Reflected Solar Image of International Commerce Centre, designed on a rising Dragon concept, taken at 15th Floor, K K Leung Building, HKU, , Hong Kong Island, at 2:45 pm, on 28 December, 2012, with the mimic solar dragon hovering onto Victoria Harbour, a world-iconic sundial surface. Photo courtesy of Mr Anthony P.C. Chung. The ICC reflected solar image Sundial on Harbour water surface is the first one discovered in the World. So ICC Sundial, the Menelaus Sundial (since ICC stands on Menelaus Point), the Silver Dragon Sundial shines on water surface, and hovers over land. Oh what another world-wonder of Menelaus Silver Dragon Sea- Land Sundial 萬年樂善銀龍海陸日晷 ! [11, 19, 20, Fig. 9, Fig. 12a, Fig. 12b, Fig. 15a-d].

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