2010 Guest Book Entries
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Name: Milo Gardner Location: Sacramento, CA Date/Time: Wednesday, December 1, 2010 12:15:11 PM Comments: Kevin, When you have time may be you can read Kevin Brown's Egyptian Fractions page: ****mathpages****home/kmath340/kmath340.htm Brown's proof that Ahmes' 1650 BCE 2/n table and 400 AD n/17 and n/19 tables were consistent with modern number theory may be new to you. Where Brown and I disagree relates to guessing one or more origins of EF arithmetic. Brown suspects that gambling was a consideration. My view is that finite conversions of infinite series 'round off' problems, were based in economics and wage payments, and was one of several origins of EF arithmetic. Thank you again for your excellent 1992 EF paper. It brings back memories of Sylvester's 1880 and 1891 papers, the 1891 paper not reporting (n/p - 1/m) = (mn -p)/mp a method that Fibonacci tried to set (mn -p)= 1. When Fibonacci could not find an LCM that calculated unity (ie. 4/13 - 1/4 = 3/52) Sylvester's suggested that Fibonacci applied a n-step "greedy algorithm". The Liber Abaci raw data shows that one additional LCM m was chosen, LCM 18 to solve 4/13 to finalize the medieval version of 3,200 year old unit fraction arithmetic. No greedy algorithm was applied by Fibonacci in a 7th distinction (Sigler's 2002 translation). In summary, we may agree medieval unit fraction math died around 1454 AD and was replaced in 1585 AD when two base 10 decimal books, one for business and one for science, were approved by the Paris Academy. Napier added to the early books, as did others, defining rigorous foundations of our base 10 decimal system. Best Regards, Milo Gardner Top 3 Hikes: 1. he Washington Monument ... taken in 1957 .. now closed |
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Name: Milo Gardner Location: Sacramento, CA Date/Time: Tuesday, November 30, 2010 4:10:58 PM Comments: Hi Kevin, From your 1992 Egyptian fraction paper was great. I enjoyed reading the Yokota Algorithm conversion: 16/17 = 1/2 + 1/3 + 1/12 + 1/51 + 1/204 and discussion of general number theory cases. My view is that modern algorithms have limitations and do not solve historical Egyptian fraction problems as well as ancient scribes worked the problems. I've worked this class of problems as long as you have ... by decoding ancient texts line by line. The "best" ancient solutions outline by a 2/n table rule Ahmes applied to solve 28/97 = 26/97 + 2/97 (RMP 31), by LCM 4, and 56, respectively, namely 26/97(4/4) = 104/(4*97) = (97 + 4 + 2 + 1)/(4*97) = 1/4 + 1/97 + 1/194 + 1/388 2/97(56/56) = 112/(56*97)= (97 + 8 + 7)/(56*97) = 1/56 + 1/679 + 1/776 and, 30/53 = 28/53 + 2/53 (RMP 36), by LCM 2 and 30. respectively, namely 28/53(2/2) = 56/106 = (53 + 2 + 1)/106 = 1/2 + 1/53 + 1/106 and 2/53 (30/30) = 60/1590 = (53 + 5 + 2)/1590 = 1/30 + 1/318 + 1/795 Let's look at Ahmes' IMPLICIT 2/n table rule: n/p = (n-2)/p + 2/p and apply it to your 16/17 discussion. Ahmes would have seen 16/17 = 14/17 + 2/17 14/17 (4/4) = 56/68 = (34 + 17 + 4 + 1)/68 = 1/2 + 1/4 + 1/17 + 1/68 2/17 (12/12) = 24/(12*17) = (17 + 4 + 3)/(12*17) = 1/12 + 1/51 + 1/68 such that: 16/17 = 1/2 + 1/4 + 1/12 + 1/17 + 1/34 + 1/51 since 1/68 + 1/68 = 1/34 That is, which 16/17 series is better? Ahmes' 1/2 + 1/4 + 1/12 + 1/17 + 1/34 + 1/51, or, Yokota's modern 1/2 + 1/3 + 1/12 + 1/51 + 1/204? Yokota's is shorter, but Ahmes offers a smaller last term by a factor of four. I'd vote for Ahmes' smaller last term. In summary Egyptian scribes converted n/p by LCM m to mn/mp inspecting divisors of mp, and choosing the best that summed to mn. Red color was added to the numerator information to add emphasis. Thank you very much for considering to extend your excellent 1992 paper to report actual RMP table construction methods. As an aside, Fibonacci used a subtraction context, converting n/p - 1/m = (mn - p)/mp, setting mn -p = 1 whenever possible. In the case of (4/13 -1/4) = (16 - 13)/52 a problem that was solved by LCM 18 such that: (3/52 - 1/18) = (54 - 52)/(18*52) = 1/468 such that, 4/13 = 1/4 + 1/18 + 1/468 a method that J.J. Sylvester mis-read in 1891 as a greedy algorithm, which is surely was not. Ahmes would have not needed two-steps, per: 4/13(4/4) = (13 + 2 + 1)/52 = 1/4 + 1/26 + 1/52 again pointing out, the older the method the better the method. Best Regards, Milo Gardner |
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Name: milo gardnner Location: Sacramento, CA Date/Time: Tuesday, November 30, 2010 1:51:10 PM Comments: Hi Kevin, Reading your 1992 Egyptian math paper: http://kevingong.com/Math/EgyptianFractions.pdf a Yokota Algorithm reported the conversion 16/17 = 1/2 + 1/3 + 1/12 + 1/51 + 1/204 and discussed 4/n, 5/n, ..., modern number theory cases in ways that ancient scribes would not have understood. My view is that modern algorithms do not solve historical Egyptian fraction problems. Ancient finite math found better concise series. The class of "best" solutions is suggested by a 2/n table rule. Ahmes applied it to solve 28/97 = 26/97 + 2/97 (RMP 31), by LCM 4, and 56, respectively, and, 30/53 = 28/53 + 2/53 (RMP 36), by LCM 2 and 30. respectively. Let's look at Ahmes' 2/n table rule: n/p = (n-2)/p + 2/p Applying Akmes rule to your 16/17 discussion. 16/17 = 14/17 + 2/17 14/17 (4/4) = 56/68 = (34 + 17 + 4 + 1)/68 = 1/2 + 1/4 + 1/17 + 1/68 2/17 (12/12) = 24/(12*17) = (17 + 4 + 3)/(12*17) = 1/12 + 1/51 + 1/68 such that: 16/17 = 1/2 + 1/4 + 1/12 + 1/17 + 1/34 + 1/51 since 1/68 + 1/68 = 1/34 That is, which 16/17 series is better: Ahmes' 1/2 + 1/4 + 1/12 + 1/17 + 1/34 + 1/51, or, your (Yokota's) 1/2 + 1/3 + 1/12 + 1/51 + 1/204? Yokota's is shorter by one term, but Ahmes offers the smaller last term by a factor of four. I'd vote for Ahmes' smaller last term using LCM m and red auxiliary numbers. Best Regards, Milo Gardner Top 3 Hikes: 1. Mt. Lassen in 1950 |
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Name: David Crose Location: Yreka/CA/USA Date/Time: , November 13, 2010 4:45:57 PM Comments: Love your page! I decided to take the long way up to Sky High Lakes in the Marble Mountains after I read your trip report. I'm glad I did (although my wife still wont let me forget the extra mileage and elevation!) as I got some great shots of the lake and Marble Valley. If you ever come back this way. check out Boulder Peak and Wright Lakes, you wont be disappointed. Top 3 Hikes: 1. PCT Parks Creek to Dunsmuir 2. King's Castle (Marble Mountains) 3. Anything in Yosemite!! |
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Name: Jim Location: Los Angeles, CA Date/Time: Saturday, September 24, 2010 8:53:31 PM Comments: I've used your site for hiking ideas for the past few years -- just wanted to say thanks! Top 3 Hikes: 1. Zion Narrows - Utah 2. Exit Glacier/Harding Icefield - Alaska 3. Eagle Creek - Oregon |
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Name: Linda Location: Rocklin CA Date/Time: Tuesday, September 14, 2010 4:27:03 PM Comments: I love your site! I'm new to hiking and find all the info here useful and I'm truly thankful! I finished my first ever 5 mile hike at Woods Lake, my boyfriend spent many years there as a young boy and he took me up the reverse way thru the camp sites and up the steep hill to Round Top lake. On the way up, I made him take my photo sitting in the old car just like on your page. What a beautiful way to spend 5 hours hiking thru the woods. We are going back in two weeks to car camp and do some more hiking. You are a true inspiration, I'm 47 and just starting to learn about hiking, I thank you for everything you have shared. Linda Top 3 Hikes: 1. ROUND TOP 2. Bonita Point Light House, San Francisco 3. Muir Woods |
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Name: Mark Location: Felton/California/United States Date/Time: Sunday, August 29, 2010 2:32:33 PM Comments: Badass website!!! I use it a lot to look up hikes both locally and far away. Your "on the trail" insight is very useful, and it's awesome how you record what you do. Keep it up! Top 3 Hikes: 1. Half Dome 2. Long's Peak (RMNP) 3. Quartz Loop (Glacier NP) |
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Name: George Location: California Date/Time: Wednesday, July 21, 2010 12:42:16 PM Comments: Congratulations on an excellent website. I must, however, come to the defense of poor Bathtub Lake in Lassen Volcanic National Park, which you savaged in your review. First point: The reason people swim in Bathtub Lake, and not nearby Butte Lake, is because Bathtub's water temperature is several degrees warmer. Second point: You wondered how the people you observed swimming on the opposite (north) shore of the lake could stand it, as you (alone on the south shore) were experiencing mud bottom and shallow water. Answer: The beach on the north side is a sandy beach that slopes into the lake, getting over your head in about 5 yards from the shore. That's why everyone (except you) was on the north shore. But, you're right about a couple of things: the NPS prescribed burn removed most of the shade. Also, the first Bathtub Lake is a working lake, i.e., its popularity is due to its providing a swimming spot for hikers and campers in the Butte Lake area, and cannot compete as scenic "alpine lake", an category in which you found it "disappointing". Top 3 Hikes: 1. Bumpass Hell, Lassen Volcanic National Park, CA 2. West Fork Trail, Oak Creek Canyon, Coconino N.F., AZ 3. Emerald Ridge, Mt. Rainier National Park, WA |
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Name: Jon Kibrick Location: Mill Valley, CA Date/Time: Sunday, July 4, 2010 10:24:01 AM Comments: You are an inspiration to the hiking community. Thank you for all the organized postings, ratings, and picture galleries. My wife and I have done many of the same hikes, and your detailed logging has inspired us to do many more of yours. |
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Name: Eric Location: Chula Vista CA Date/Time: Thursday, July 1, 2010 4:11:56 PM Comments: Great website! I wandered into it while googling Lembert Dome. Your photographs are beautiful. Thank you for creating this site. Top 3 Hikes: 1. Tuolumne to Red's Meadow 2. Angel's Landing in Zion 3. Whitney/Tallac/Half Dome |
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Name: Julie Location: Santa Clarita, CA Date/Time: Thursday, June 17, 2010 4:08:56 PM Comments: I just LOVE your website and it is so kind of you to share with us all...I used it years ago to plan a Sequoia trip and just happened across it again and you have Lassen NP which is where we are going in July. Your pictures make it SO real I can't wait to go. Thank you again! Top 3 Hikes: 1. Paradise Valley Ca- back packing 2. River walking in Zion 3. Tuolumne Meadows Yosemite |
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Name: Susan Location: Highland Park, NJ Date/Time: Sunday, June 13, 2010 6:30:14 AM Comments: Hey! I really like your site. I found it because we are getting ready to take our 4 year old for her first camping trip and your tips are great. We're all looking forward to it. Our older daughter was an avid hiker and I'm hoping the little one will be too. Top 3 Hikes: 1. Cinque Terra, Italy 2. Wind Cave,South Dakota 3. Assateague Island, Maryland |
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Name: Lynette H. Location: Lodi, CA Date/Time: Thursday, June 10, 2010 9:08:23 AM Comments: Nice site...great information! There are quite a few hikes listed that I have done and many that I look forward to hiking. |
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Name: Angela Location: Houston/TX/USA Date/Time: Sunday, May 16, 2010 10:15:57 PM Comments: Thank you very much for putting up this hiking site. It is really helpful to have the photos and descriptions of the hikes. Very user-friendly and visual. Better than the Moon Handbook I got! |
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Name: Tracy Kim Location: Fremont, California Date/Time: Thursday, May 6, 2010 8:58:26 AM Comments: Hello Kevin, My husband and I recently became "hike lovers" :-) and we've been enjoying it so much ever since.. exploring Bay Area in such a different approach, it's absolutely amazing. My husband and I wanted to thank you!! for such a helpful website you put together. We are going to Kalalau Trail in Kauai next week, so excited !! |
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Name: Sam Son Location: Ranktown, KY, Jelly Date/Time: Wednesday, April 14, 2010 9:00:15 PM Comments: I fell from the top of Moro Rock. Cracked a smile all the way down. Knocked some sense into me, I'll tell ya that. Can't wait to do the wheelchair loop on Mont Blanc in December! Top 3 Hikes: 1. Black's Beach, San Diego 2. Matterhorn, Disneyland 3. Magic Mountain, Valencia |
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Name: Vicky Location: San Francisco, CA Date/Time: Monday, April 12, 2010 4:02:33 PM Comments: Thanks for building and sharing this wonderful site. I plan to go camping and hiking in Sonora Pass area or Kings Canyon in July. Love your 6 days backpacking in Kings Canyon!! Top 3 Hikes: 1. Mt Whitney 2. Rae Lakes Loop 3. Half Dome |
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Name: Julie Location: South Lake Tahoe, CA, USA Date/Time: Sunday, April 11, 2010 9:42:43 AM Comments: Thank you for sharing your experiences. Visiting your site has been quite fun. So many trails; so little time. Thank you. |
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Name: Don Location: Littleton, CO Date/Time: Tuesday, March 2, 2010 3:58:04 PM Comments: Fun and interesting web site! May I suggest another couple of resources for your list: Roughing It Easy by Dian Thomas, a great guide for family camping (although some concepts may be a little outdated due to leave-no-trace camping; and the Boy Scout Handbook. Top 3 Hikes: 1. Lost Creek, near Bailey CO 2. All hikes in Yosemite |
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Name: Chris Location: Healdsburg CA Date/Time: Wednesday, January 13, 2010 3:55:40 PM Comments: Nice site and good info. thanks! As a youth I came across a list of the 100 books to read. I have compiled the 100 hikes to enjoy. Many you offer are there. Keep up the good work and hope to see you on the trail. Top 3 Hikes: 1. Long Beach on Vancouver Island 2. JMT in 1979 3. Anything with Jeannie. |
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Name: Fiona G Location: UK Date/Time: Wednesday, January 6, 2010 8:58:54 AM Comments: Hi Kevin, thanks so much for your website. We're visiting family in SoCal this coming summer and are planning a trip to Sequoia as part of the tour - so I'm finding your pics and info really helpful. Thanks for sharing! Fiona. |
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Name: Brendan Location: Atherton, CA Date/Time: Monday, January 4, 2010 12:57:58 PM Comments: Kevin, Many thanks for the great info for hiking on the Peninsula. This is a terrific public resource in the true spirit of hiking and fair use of the internet. Brendan |
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