Sigtuna Market Fair 2011

besökte Sveriges första och kanske mysigaste stad i helgen. (Aug 28, 2011)

▶ Slideshow:
Sigtuna Möte - en marknad i 1912 års anda

Rachmaninoff concerto 1

Rachmaninoff piano concerto No. 1 in F-sharp minor, Op. 1- Ashkenazy-Previn-Tempo 2

Rachmaninoff composed this Concerto at age 19. I personally find it his most precious piece, if not his most sophisticated composition. Like an unbridled confession, it touches my unsettled self deep down inside. Every time I listen to this, I think of a passage in Boris Vian's novel L'Écume des jours (1947). Exactly why I don't know ...

"À l'endroit où les fleuves se jettent dans la mer, il se forme une barre difficile à franchir et de grands remous écumeux où dansent les épaves. Entre la nuit du dehors et la lumière de la lampe, les souvenirs refluaient de l'obscurité, se heurtaient à la clarté et, tantôt apparents, montraient leurs ventres blancs et leurs dos argentés."

"Before I think where I go"

"Far, far in the forest, before I think where I go,
Solitary, smelling the earthy smell, stopping now and then in the silence,
Alone I had thought —"

From Walt Whitman (1819–1892) Leaves of Grass (1900)

Stockholm i mitt hjärta

Glad midsommar!

Another midsummer night's dream:
photobucket.com/florae

Midsummer in Husby

- Eritrean twin girls celebrating Midsummer in Husby, a suburban area of Stockholm mainly inhabited by immigrants.

I didn't pose them, but upon hearing my request to take a photo, one of them rested her hand on the shoulder of the other girl, with a simple smile that appeared to me most disarming. So everything went on naturally. I was fortunate to witness this unguarded innocence. Just beautiful.

Après le cubisme

Après le cubisme by Ozenfant

Never a fan of the majority of Le Corbusier's creative works, I personally find him to be greater in articulating rhetoric than practicing architecture. But still, always powerful rhetoric, admittedly... This book is but one example, it almost reads like a manifesto for the emerging spirit of a modern man. So after I finished the book, though it didn't at once convert me to his grand visions, it certainly left me considerably charmed.



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World's Second-Hardest Easy Geometry Problem

This is my solution in response to "World's Second-Hardest Easy Geometry Problem".

Henceforth I use symbol ≅ to indicate triangle congruence.

Lemma:

In graph 1, suppose that triangle ΔIEH is equilateral, i.e. IE=EH=IH, and if ∠DIH=∠DHI=40°, then we conclude line DE bisects the angle ∠IEH, i.e., ∠DEI=∠DEH=(1/2)*∠IEH=(1/2)*60°=30°.

Proof: From point D, draw an altitude of the triangle ΔDIH which intersects line IH at point O. Since ∠DIH=∠DHI, and ∠DOI=∠DOH=90°, DO=DO, therefore triangle ΔDOI ≅ triangle ΔDOH, so IO=OH. Connects points O and E with a line OE, because IO=OH, OE=OE, IE=EH, therefore triangle ΔIOE ≅ triangle ΔHOE, so ∠IOE=∠HOE. Since ∠IOE+∠HOE=180°, it follows that ∠IOE=∠HOE=90°. So ∠DOH+∠HOE=180°, therefore points D, O, E lie on a straight line. Since line OE bisects the angle ∠IEH, we conclude that the same line DF bisects the angle ∠IEH, too.

Now back to the original problem. Draw a line HB counter-clockwise 20 degrees from line CB and line HB intersects line AE at point I, ∠IBA=∠IBD+∠DBA=∠CBD-∠CBI+∠DBA=30°-20°+50°=60°. Since∠IAB=60°, so triangle ΔIAB is equilateral, AB=AI=BI. Since ∠CAB=∠CBA=80°, triangle ΔCAB is isosceles, so AC=BC, ∠CAE=∠HBC=20°, ∠C=∠C, therefore triangle ΔCAE ≅ triangle ΔCBH, so CH=CE, triangle ΔCHE is also isosceles, line HE is parallel to line AB, ∠EHI=∠HEI=60°, hence triangle ΔHIE is also equilateral, therefore ∠HIE=60°.

Since ∠DAB=80°, ∠DBA=50°, ∠BDA=180°-80°-50°=50°, so triangle ΔADB is isosceles, AD=AB, but AB=AI, so AD=AI, therefore triangle ΔADI is isosceles, too. ∠DIA=(1/2)*(180°-∠DAI)=(1/2)*(180°-20°)=80°

Therefore ∠HID=180°-∠HIE-∠DIA=180°-60°-80°=40°

∠AHB=180°-∠HAB-∠HBA=180°-80°-60°=40°, i.e., ∠DHI=40°=∠HID, and we know triangle ΔHIE is equilateral. From the lemma we proved in the beginning, we conclude that line DE bisects the angle ∠AEH, i.e., ∠DEA=30°


Thus angle x is equal to 30 degrees.

World's Hardest Easy Geometry Problem

This is my solution in response to "World's Hardest Easy Geometry Problem".

Henceforth I use symbol ~ to denote being "similar" between two triangles; symbol ≅ to indicate triangle congruence.

Draw a line that bisects the angle ∠CBD - let it intersect line AE at point I and line AC at point P, so ∠CBP=∠PBD=(1/2)*20°=10°. Draw another line that is 10 degrees down clockwise from line AE and intersects line BD at point H. Because ∠IAB=70°, ∠IAH=10°, so ∠HAB=70°-10°=60°, and because ∠HBA=60°, so ∠BHA=180°-60°-60°=60°. Therefore triangle ΔABH is equilateral, meaning that AH=HB=AB.

Draw a line between point D and I. Because ∠DAI=∠IBD(=10°), it follows that the four points D, I, B, A lie on a circle, therefore the inscribed angle ∠IDB=∠IAB=70°, and the inscribed angle ∠DIA=∠DBA=60°. Line PDA and Line PIB are intersecting circle secants, therefore triangle ΔPDI ~ triangle ΔPBA, ∠PID=∠PAB=80°; ∠PDI=∠PBA=70°

and PI/DI=AP/AB (1)

Because ∠CAB=∠CBA=80° triangle ΔCAB is isosceles, CA=CB, likewise because ∠IAB=∠IBA=70° triangle ΔIAB is isosceles, IA=IB. Draw a line that connects points I and H, because AI=IB, and AH=HB, so triangle ΔAIH ≅ triangle ΔBIH, so ∠AIH=∠BIH=(1/2)*∠AIB=(1/2)*(180°-70°-70°)=20°. Draw a line that connects points C and I, because CA=CB, IA=IB, CI=CI, so triangle ΔCAI ≅ triangle ΔCBI, so ∠ACI=∠BCI=(1/2)*∠ACB=(1/2)*(180°-80°-80°)=10°. By symmetry both points I and H lie on the same line that bisects the angle ∠ACB. Also because ∠CAE=∠CBP=10°, ∠ACB=∠BCA, CA=CB, therefore triangle ΔCAE ≅ triangle ΔCPB, so PB=AE, since IA=IB, so PB-IB=AE=AI, ie, PI=EI. Then ∠PCI=∠ECI, PC=CE, CI=CI, so triangle ΔPCI ≅ triangle ΔECI, ∠CIP=∠CIE=∠APB-∠ACI=∠ACB+∠CBP-∠ACI=20°+10°-10°=20°.

Since ∠PDI=70°=∠IDH. Because ∠PID=80° and ∠HID=∠AIH+∠DIA=20°+60°=80°, so ∠PID=∠HID, DI=ID, so triangle ΔPDI ≅ triangle ΔHDI, therefore IH=IP=IE. Further, since ∠PCI=∠IBH=10°, ∠CIP=∠BIH=20°, and IP=IH, therefore triangle ΔCPI ≅ triangle ΔBHI so CE=CP=HB=HA=AB

Since triangle ΔPDI ~ triangle ΔPBA, and triangle ΔPDI ≅ triangle ΔHDI, so triangle ΔHDI and triangle ΔPBA would be similar as well,

so HI/DH=AP/PB
in other words, AP*DH=PB*HI (2)

Because ∠PCB=∠HIB=20°, and ∠CBP=∠IBH=10°, so triangle ΔPCB ~ triangle ΔHIB,
It follows that PB/PC=HB/HI, therefore PB*HI=CP*HB=HA*AB (3)

Compare (2) and (3) we have AP*DH=HA*AB, so HA/DH=AP/AB, but according to earlier conclusion (1), we know that AP/AB=PI/DI=EI/DI,

Therefore HA/DH=EI/DI
Since ∠DHI=180°-60°=120°, ∠DIE=∠DIP+∠CIP+∠CIE=80°+20°+20°=120°, so ∠DHI=∠DIE

Therefore triangle ΔDAH ~ triangle ΔDEI

Therefore ∠DEI=∠DAH=∠PAI+∠IAH=10°+10°=20°


Thus, angle x is equal to 20 degrees.

Horisont

Det fanns en tid som var varken vinter eller vår. Tiden, som bara hade strömmade ur de gamla tankarna, stod still denna isländska natt i början av maj månad. Under en mycket klar himmel troddes temperaturen utomhus ligga nära fryspunkten. När man kastade en blick åt blånande berg vid öppna havet längs horisonten, plötsligt kändes ens egen existens futtig och osäker. Sen började man erinra sig en världsbild vars innehåll berättade att allting kunde ju inte bli färdigt hädanefter. Den tiden försvann då, men jag fortfarande har dessa tankar i minnet.

Nu saknar jag Island ändå.

Come Undone

Fragment

Go green

What do people think of Earth Day, now that it's been a yearly ritual all over the world? Living in an environment-conscious country, when I go to a local fast food chain MAXI, I'm reminded by the burger menu how much CO2 emission would be associated with each meal. I think it's a better call this way than putting forward all the PR-stuffs and diverting the attention from the environment impact of their core product, which is food. On the other hand, isn't the fact that, in term of the amount of greenhouse gases caused by its production, meat ranks the worst offender in all types of food, posing an inherent challenge to a burger restaurant, just like forcing the Tobacco companies to add a health hazard note to each cigarette pack sold would do?




Also, remember the Earth Hour? In 2009 the dairy maker Arla had started a public campaign in which their milk packages were painted all black to raise the awareness of the EH message that lights be switched off for one hour on the event day. What I found interesting was what happened after the campaign just ended, and here's one visual testimony in a grocery store the day after:




I'm supporting all the green initiatives, just people need to understand it's not about merely wearing politically correct tags and feeling good but our everyday lifestyle that can make a real difference to tackle the environmental challenges the world is facing. Like now, in springtime, I believe every day is an Earth Day, and everyone can be a TreeHugger. These are my Easter prayers.




P.S. In case you'll take a ride when visiting Stockholm in the summer, aside from taking more rented bikes for transportation, you may wish to try the Eco "Car" that has been introduced to the Swedish capital:

FxCamera

TechCrunch reported this week that, on Flickr, Apple’s iPhone 4 is set to overtake Nikon D90 and become the most popular camera used. I'm an Android user, the only thing I envy about the iPhone is its Hipstamatic app which I find great fun to play with. But today I finally found an Android app that could do a similar job as Hipstamatic: FxCamera. It's a free product from Android Market, so worth a try. Here is my first phone camera shot processed by this software. -> Check out the complete gallery: pbase.com/tsienni/mobile